Irenaeus gives the following account of the doctrines of Basilides in Against Heresies, Book I, Chapter 24, sections 3-7.
3. Basilides again, that he may appear to have discovered something more sublime and plausible, gives an immense development to his doctrines. He sets forth that Nous was first born of the unborn father, that from him, again, was born Logos, from Logos Phronesis, from Phronesis Sophia and Dynamis, and from Dynamis and Sophia the powers, and principalities, and angels, whom he also calls the first; and that by them the first heaven was made. Then other powers, being formed by emanation from these, created another heaven similar to the first; and in like manner, when others, again, had been formed by emanation from them, corresponding exactly to those above them, these, too, framed another third heaven; and then from this third, in downward order, there was a fourth succession of descendants; and so on, after the same fashion, they declare that more and more principalities and angels were formed, and three hundred and sixty-five heavens. Wherefore the year contains the same number of days in conformity with the number of the heavens.
4. Those angels who occupy the lowest heaven, that, namely, which is visible to us, formed all the things which are in the world, and made allotments among themselves of the earth and of those nations which are upon it. The chief of them is he who is thought to be the God of the Jews; and inasmuch as he desired to render the other nations subject to his own people, that is, the Jews, all the other princes resisted and opposed him. Wherefore all other nations were at enmity with his nation. But the father without birth and without name, perceiving that they would be destroyed, sent his own first-begotten Nous (he it is who is called Christ) to bestow deliverance on them that believe in him, from the power of those who made the world. He appeared, then, on earth as a man, to the nations of these powers, and wrought miracles. Wherefore he did not himself suffer death, but Simon, a certain man of Cyrene, being compelled, bore the cross in his stead; so that this latter being transfigured by him, that he might be thought to be Jesus, was crucified, through ignorance and error, while Jesus himself received the form of Simon, and, standing by, laughed at them. For since he was an incorporeal power, and the Nous (mind) of the unborn father, he transfigured himself as he pleased, and thus ascended to him who had sent him, deriding them, inasmuch as he could not be laid hold of, and was invisible to all. Those, then, who know these things have been freed from the principalities who formed the world; so that it is not incumbent on us to confess him who was crucified, but him who came in the form of a man, and was thought to be crucified, and was called Jesus, and was sent by the father, that by this dispensation he might destroy the works of the makers of the world. If any one, therefore, he declares, confesses the crucified, that man is still a slave, and under the power of those who formed our bodies; but he who denies him has been freed from these beings, and is acquainted with the dispensation of the unborn father.
5. Salvation belongs to the soul alone, for the body is by nature subject to corruption. He declares, too, that the prophecies were derived from those powers who were the makers of the world, but the law was specially given by their chief, who led the people out of the land of Egypt. He attaches no importance to [the question regarding] meats offered in sacrifice to idols, thinks them of no consequence, and makes use of them without any hesitation; he holds also the use of other things, and the practice of every kind of lust, a matter of perfect indifference. These men, moreover, practise magic; and use images, incantations, invocations, and every other kind of curious art. Coining also certain names as if they were those of the angels, they proclaim some of these as belonging to the first, and others to the second heaven; and then they strive to set forth the names, principles, angels, and powers of the three hundred and sixty-five imagined heavens. They also affirm that the barbarous name in which the Saviour ascended and descended, is Caulacau.
6. He, then, who has learned [these things], and known all the angels and their causes, is rendered invisible and incomprehensible to the angels and all the powers, even as Caulacau also was. And as the son was unknown to all, so must they also be known by no one; but while they know all, and pass through all, they themselves remain invisible and unknown to all; for, "Do thou," they say, "know all, but let nobody know thee." For this reason, persons of such a persuasion are also ready to recant [their opinions], yea, rather, it is impossible that they should suffer on account of a mere name, since they are like to all. The multitude, however, cannot understand these matters, but only one out of a thousand, or two out of ten thousand. They declare that they are no longer Jews, and that they are not yet Christians; and that it is not at all fitting to speak openly of their mysteries, but right to keep them secret by preserving silence.
7. They make out the local position of the three hundred and sixty-five heavens in the same way as do mathematicians. For, accepting the theorems of these latter, they have transferred them to their own type of doctrine. They hold that their chief is Abraxas; and, on this account, that word contains in itself the numbers amounting to three hundred and sixty-five.
Irenaeus also makes this accusation (op. cit., I.28.2).
2. Others, again, following upon Basilides and Carpocrates, have introduced promiscuous intercourse and a plurality of wives, and are indifferent about eating meats sacrificed to idols, maintaining that God does not greatly regard such matters. But why continue? For it is an impracticable attempt to mention all those who, in one way or another, have fallen away from the truth.
Irenaeus also makes mention of Basilides as follows (op. cit., II.2.3).
3. If, however, [the things referred to were done] not against His will, but with His concurrence and knowledge, as some [of these men] think, the angels, or the Former of the world [whoever that may have been], will no longer be the causes of that formation, but the will of God. For if He is the Former of the world, He too made the angels, or at least was the cause of their creation; and He will be regarded as having made the world who prepared the causes of its formation. Although they maintain that the angels were made by a long succession downwards, or that the Former of the world [sprang] from the Supreme Father, as Basilides asserts; nevertheless that which is the cause of those things which have been made will still be traced to Him who was the Author of such a succession. [The case stands] just as regards success in war, which is ascribed to the king who prepared those things which are the cause of victory; and, in like manner, the creation of any state, or of any work, is referred to him who prepared materials for the accomplishment of those results which were afterwards brought about.
Irenaeus again makes mention of Basilides (op. cit., II.13.8).
Now, these remarks which have been made concerning the emission of intelligence are in like manner applicable in opposition to those who belong to the school of Basilides, as well as in opposition to the rest of the Gnostics, from whom these also (the Valentinians) have adopted the ideas about emissions, and were refuted in the first book.
Again Irenaeus makes mention of Basilides (op. cit., II.16.2).
This difficulty presented itself to Basilides after he had utterly missed the truth, and was conceiving that, by an infinite succession of those beings that were formed from one another, he might escape such perplexity. When he had proclaimed that three hundred and sixty-five heavens were formed through succession and similitude by one another, and that a manifest proof [of the existence] of these was found in the number of the days of the year, as I stated before; and that above these there was a power which they also style Unnameable, and its dispensation--he did not even in this way escape such perplexity. For, when asked whence came the image of its configuration to that heaven which is above all, and from which he wishes the rest to be regarded as having been formed by means of succession, he will say, from that dispensation which belongs to the Unnameable. He must then say, either that the Unspeakable formed it of himself, or he will find it necessary to acknowledge that there is some other power above this being, from whom his unnameable One derived such vast numbers of configurations as do, according to him, exist.
Irenaeus mentions Basilides again (op. cit., II.16.4).
4. As to the accusation brought against us by the followers of Valentinus, when they declare that we continue in that Hebdomad which is below, as if we could not lift our minds on high, nor understand those things which are above, because we do not accept their monstrous assertions: this very charge do the followers of Basilides bring in turn against them, inasmuch as they (the Valentinians) keep circling about those things which are below, [going] as far as the first and second Ogdoad, and because they unskilfully imagine that, immediately after the thirty AEons, they have discovered Him who is above all things Father, not following out in thought their investigations to that Pleroma which is above the three hundred and sixty-five heavens, which is above forty-five Ogdoads. And any one, again, might bring against them the same charge, by imagining four thousand three hundred and eighty heavens, or AEons, since the days of the year contain that number of hours. If, again, some one adds also the nights, thus doubling the hours which have been mentioned, imagining that [in this way] he has discovered a great multitude of Ogdoads, and a kind of innumerable company of AEons, and thus, in opposition to Him who is above all things Father, conceiving himself more perfect than all [others], he will bring the same charge against all, inasmuch as they are not capable of rising to the conception of such a multitude of heavens or AEons as he has announced, but are either so deficient as to remain among those things which are below, or continue in the intermediate space.
Irenaeus again refers to Basilides in his refutation (op. cit., II.35.1).
Moreover, in addition to what has been said, Basilides himself will, according to his own principles, find it necessary to maintain not only that there are three hundred and sixty-five heavens made in succession by one another, but that an immense and innumerable multitude of heavens have always been in the process of being made, and are being made, and will continue to be made, so that the formation of heavens of this kind can never cease. For if from the efflux(5) of the first heaven the second was made after its likeness, and the third after the likeness of the second, and so on with all the remaining subsequent ones, then it follows, as a matter of necessity, that from the efflux of our heaven, which he indeed terms the last, another be formed like to it, and from that again a third; and thus there can never cease, either the process of efflux from those heavens which have been already made, or the manufacture of [new] heavens, but the operation must go on ad infinitum, and give rise to a number of heavens which will be altogether indefinite.
And Irenaeus, in a few other passages, groups Basilides along with other gnostics such as Marcion, Valentinus, and Carpocrates.
Paul Allan Mirecki writes on the attestation of Clement of Alexandria to Basilides (The Anchor Bible Dictionary, v. 1, p. 624):
Clement of Alexandria (ca. 200) quotes and comments on seven sections from unknown works of Basilides in Clement's famous Stromata 4-5 (cf. frags. A through E, G and H in Layton 1987:427-37, 440-44). Two of the fragments focus on cosmological issues (Layton 1987:428-31): the first is a quote (frag. A; Strom. 4.162.1) which refers to two of the constituent members of the octet in the godhead and is in agreement with Irenaeus' description (Haer. 1.24.3), while the other is a quote and discussion (frag. B; Strom. 5.74.3) refering to the Stoic concept of the uniqueness of the world. The five remaining fragments focus on ethical issues (in line with Stoic ethics; cf. Layton 1987: 418, 432-44): the first (frag. C; Strom. 5.3.2-3) refers to Basilides' teaching on election in relation to faith and virtue; the second (frag. D; frag. 4 in Volker; Strom. 4.86.1) describes Basilides' teaching on the will of God (fate) to which the virtuous person aspires; the third (=frag. E; Strom. 4.165.3) describes Basilides' teaching that human souls retain their identity through their various incarnations and so transcend the world; the fourth (frag. G; frag. 2 in Volker; Strom. 4.81.2-4.83.2) is a series of lengthy quotes introduced by Clement with the statement that they are from Book 23 of Basilides' now lost Commentaries (the Exegetica), quotes which seem to be from a commentary on 1 Pet 4:12-19 in which he argues that the will of god (fate) is "all-powerful and all good" (Layton 1987: 440-43); the fifth (frag. H; Strom. 4.153.3) is a single sentnce referring to forgivable sins.
Mirecki also writes about the connection of Origen to Basilides (op. cit., p. 624):
Origen, in his commentary on Romans (ca. 244), quotes a Basilidean text (frag. F; frag. 3 in Volker; Origenes, Opera Omnia 4) in which Basilides interprets Rom 7:7 to refer to incarnation (which Origen accepted) and a certain gnostic cosmology (which Origen rejected). this text may also come from Basilides' Commentaries like frag. G in Clement, suggesting that Basilides and the eastern Valentinian gnostic Heracleon (ca. 150; Rudolph 1977:323-24) are the first known authors of commentaries on NT texts.
And Mirecki observes on the other witnesses to Basilides as follows (op. cit., p. 624):
Other early and descriptive patristic refutations of Basilides' teachings are known. One is found in an extant herisiology by Hippolytus of Rome (first half of the 3d century; Haer. 7:20-27). But Hippolytus' report is not in tandem with the descriptions and quotations of Basilides' system which we find in the summaries and quotations in Irenaeus, Clement of Alexandria, and Origen (see discussion in Layton 1987:418 n. 2; Rudolph 1977:310). Another is the now lost Refutation of Basilides by the heresiologist Agrippa Castor (ca. 135) mentioned by Eusebius (Hist. Eccl. 4.7.6-8) at the beginning of the 4th century (Rudolph 1977:309-10).
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Kirby, Peter. "Basilides." Early Christian Writings. <http://www.earlychristianwritings.com/basilides.html>.
