Which doctrine was formalised first: Infant Baptism or Original Sin?

Given that the earliest record after Scripture is from Justin Martyr CE 150 who argues for an informed baptism and not for infant baptism, then the answer is the idea of original sin and its implications came first and then the new idea of infant baptism took shape.

And for this [rite] we have learned from the apostles this reason. Since at our birth we were born without our own knowledge or choice, by our parents coming together, and were brought up in bad habits and wicked training; in order that we may not remain the children of necessity and of ignorance, but may become the children of choice and knowledge, and may obtain in the water the remission of sins formerly committed, there is pronounced over him who chooses to be born again, and has repented of his sins, the name of God the Father and Lord of the universe; he who leads to the laver the person that is to be washed calling him by this name alone. The First Apology Chapter LXI

Tertullian CE 200 is sometimes cited as supporting infant baptism, which if true would arise before Augustine's doctrine of original sin. But Tertullian does not even teach the necessity of water baptism for salvation or the necessity of infant baptism.

And so, according to the circumstances and disposition, and even age, of each individual, the delay of baptism is preferable; principally, however, in the case of little children. For why is it necessary—if (baptism itself) is not so necessary—that the sponsors likewise should be thrust into danger? Ethical, On Baptism

These two baptisms [water and blood] He [Christ] sent out from the wound in His pierced side, in order that they who believed in His blood might be bathed with the water; they who had been bathed in the water might likewise drink the blood. This is the baptism which both stands in lieu of the fontal bathing when that has not been received, and restores it when lost. -ibid-

Though most don't like or understand Paul's teaching, Scripture, however, informs that we are all born dead spiritually in Adam. We must be born again. But Paul says this as regards children.

For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband: else were your children unclean; but now are they holy [. 1 Cor. 7:14

And of course Christ says this about children.

But Jesus said, Suffer little children, and forbid them not, to come unto me: for of such is the kingdom of heaven. Mat. 19:14

As for the idea of original sin, most commentators point to Augustine CE 400 as the source. Some will point to Cyprian CE 200.

So, the practice of infant baptism does not enter the thinking until after the idea of original sin is developed.

The Early Church did not teach original Sin in the way that Augustine did (That we are doomed because of the Sin of Adam unless we are baptized).

Infant baptism is just the recognition that humanity needs a savior due to the Fall (but the will of humanity was not taught to be doomed in the Augustinian sense); I'm going to quote some fathers from an Orthodox article that shows Christianity had a much different position before Augustine, on "Original Sin" and then follow by a few pre-Augustinian Fathers on infant baptism, to demonstrate that the notion of infant Baptism occurs before Augustine.

But before I do, it should be noted that there are a few things going on that muddy the waters, historically speaking. The biggest being that Soteriology notions were still being ironed out in the Church in regards to the sacraments. People were not entirely certain how the issue of Sin, post baptism was dealt with, so some people like Emperor Constantine deliberately put off their baptism until they knew they were close to death. Attempting to use it a bit like a Last Rights. (the process of Sacramental confession was under development at this time).

But Anyway, here are a few quotes that refute the Augustinian outlook on Original Sin by Early Church Fathers.

Ignatius of Antioch, 35-107 AD Bishop of Antioch in Syria. A disciple of the Apostle John and appointed as Bishop of Antioch by the Apostle Peter.

I do not mean to say that there are two different human natures, but all humanity is made the same, sometimes belonging to God and sometimes to the devil. If anyone is truly spiritual they are a person of God; but if they are irreligious and not spiritual then they are a person of the devil, made such NOT by nature, but by their own choice. (The Epistle of Ignatius to the Magnesians chap 5, + Pg.61 vol. 1) There is set before us life upon our observance [of God’s precepts], but death as the result of disobedience, and every one, according to the choice he makes, shall go to his own place, let us flee from death, and make choice of life. (The Epistle of Ignatius to the Magnesians chap 5)

Irenaeus of Lyon 120-202 AD. The Apostle John had a disciple named Polycarp, who had a disciple named Irenaeus.

Those who do not do it [good] will receive the just judgment of God, because they had not worked good when they had it in their power to do so. But if some had been made by nature bad, and others good, these latter would not be deserving of praise for being good, for they were created that way, nor would the former be reprehensible, for that is how they were made. However, all men are of the same nature. They are all able to hold fast and to go what is good. On the other hand, they have the power to cast good from them and not to do it. (Against Heresies (Book IV, Chapter 37)

Justin Martyr, 110-165 AD

But neither do we affirm that it is by fate that men do what they do, or suffer what they suffer, but that each man by free choice acts rightly or sins; and that it is by the influence of the wicked demons that earnest men, such as Socrates and the like, suffer persecution and are in bonds, while Sardanapalus, Epicurus, and the like, seem to be blessed in abundance and glory. The Stoics, not observing this, maintained that all things take place according to the necessity of fate. But since God in the beginning made the race of angels and men with free-will, they will justly suffer in eternal fire the punishment of whatever sins they have committed. And this is the nature of all that is made, to be capable of vice and virtue. For neither would any of them be praiseworthy unless there were power to turn to both [virtue and vice]. And this also is shown by those men everywhere who have made laws and philosophized according to right reason, by their prescribing to do some things and refrain from others. Even the Stoic philosophers, in their doctrine of morals, steadily honour the same things, so that it is evident that they are not very felicitous in what they say about principles and incorporeal things. For if they say that human actions come to pass by fate, they will maintain either that God is nothing else than the things which are ever turning, and altering, and dissolving into the same things, and will appear to have had a comprehension only of things that are destructible, and to have looked on God Himself as emerging both in part and in whole in every wickedness; or that neither vice nor virtue is anything; which is contrary to every sound idea, reason, and sense.” /Apology 2 Ch.7 2 (+ The Anti-Nicene Fathers, Vol. I, p.354)

Clement 2nd, 80-140 AD.

Thus although we are born neither good nor bad, we become on or the other and having formed habits, we are with difficulty drawn from them. Pg 273 vol.8 He who is good by his own choice is really good; but he who is made good by another under necessity is not really good, because he is not what he is by his own choice… So, brothers and sisters, if we have done the will of the Father and have kept the flesh pure and have observed the commandments of the Lord, we will receive eternal life (2 Clement 8:4)

And there are many more.

Quotes from the old church fathers where they deny original sin / sinful nature

Quotes From Early Fathers on Baptism

Irenaeus, Bishop of Lyon

“For He came to save all through means of Himself–all, I say, who through Him are born again to God–infants, and children, and boys, and youths, and old men.” Irenaeus, Against Heresies, 2,22:4 (A.D. 180), in ANF, I:391

Hippolytus, Bishop of Rome

“And they shall baptize the little children first. And if they can answer for themselves, let them answer. But if they cannot, let their parents answer or someone from their family.” Hippolytus of Rome, Apostolic Tradition, 21(c. A.D. 215), in AT,33

Origin

“[T]herefore children are also baptized.” Origen, Homily on Luke, XIV (A.D. 233), in JER, 65 “For this reason, moreover, the Church received from the apostles the tradition of baptizing infants too.” Origen, Homily on Romans, V:9 (A.D. 244), in JER, 65 “Baptism is given for the remission of sins; and according to the usage of the Church, Baptism is given even to infants. And indeed if there were nothing in infants which required a remission of sins and nothing in them pertinent to forgiveness, the grace of baptism would seem superfluous.” Origen, Homily on Leviticus, 8:3 (post A.D. 244), in JUR, I: 208 [After quoting Leviticus 12:8 and Psalm 51:5] For this also the church had a tradition from the apostles, to give baptism even to infants. For they to whom the secrets of the divine mysteries were given knew that there is in all persons the natural stains of sin which must be washed away by the water and the Spirit. On account of these stains the body itself is called the body of sin. Origin, Commentary on Romans 5:9 (from: http://www.orlutheran.com/html/baptevid.html)

Cyprian, Bishop of Carthage

“But in respect of the case of the infants, which you say ought not to be baptized within the second or third day after their birth, and that the law of ancient circumcision should be regarded, so that you think one who is just born should not be baptized and sanctified within the eighth day…And therefore, dearest brother, this was our opinion in council, that by us no one ought to be hindered from baptism…we think is to be even more observed in respect of infants and newly-born persons..” Cyprian, To Fidus, Epistle 58(64): 2,6 (A.D. 251), in ANF, 5:353-354

[http://www.wenorthodox.com/2012/12/the-early-church-fathers-on-infant-baptism/)

Was Infant Baptism practiced before the development of the idea of Original Sin?

Ultimately it is not possible to historically know for with any great amount of surety as the two seem to be based on very little historical documents as well as the interpretation on a few choice passages of Sacred Scripture. It does seem that the two developed at the same time. There is a definite relationship between infant baptism and original sin.

The Catholic Church believes that there is a biblical basis for baptizing infants to cleanse them for original sin.

And Peter said to them, "Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you shall receive the gift of the Holy Spirit. For the promise is to you and to your children and to all that are far off, every one whom the Lord our God calls to him." And he testified with many other words and exhorted them, saying, "Save yourselves from this crooked generation." [Acts 2:38–40]

We were buried therefore with him by baptism into death, so that as Christ was raised from the dead by the glory of the Father, we too might walk in newness of life. [Romans 6:4]

God's patience waited in the days of Noah, during the building of the ark, in which a few, that is, eight persons, were saved through water. Baptism, which corresponds to this, now saves you, not as a removal of dirt from the body but as an appeal to God for a clear conscience, through the resurrection of Jesus Christ. [1 Peter 3:20–21]

The Catechism holds that infant baptism is "an immemorial tradition", practised from the earliest days of the Church recorded in Acts.

1252 The practice of infant Baptism is an immemorial tradition of the Church. There is explicit testimony to this practice from the second century on, and it is quite possible that, from the beginning of the apostolic preaching, when whole "households" received baptism, infants may also have been baptized [Cf Acts 16:15, 33; Acts 18:8; 1 Cor 1:16] - What is the Biblical basis for the Catholic teaching that infant baptism cleanses from original sin?

"Seek forgiveness" goes on to add this to this very question.

Furthermore Paul notes that baptism has replaced circumcision.

In Col. 2:11: In him you also were circumcised – not, however, with a circumcision performed by human hands, but by the removal of the fleshly body, that is, through the circumcision done by Christ. 2:12 Having been buried with him in baptism, you also have been raised with him through your faith in the power of God who raised him from the dead.

If Paul meant to exclude infants, he would not have chosen circumcision as a parallel for baptism.

Augustine also taught that,

“It is this one Spirit who makes it possible for an infant to be regenerated ... when that infant is brought to baptism; and it is through this one Spirit that the infant so presented is reborn. For it is not written, ‘Unless a man be born again by the will of his parents’ or “ by the faith of those born again of water and the Holy Spirit’[John3:5]. The water, therefore manifesting exteriorly the sacrament of grace, and the Spirit effecting interiorly the benefit of grace, both regenerate in one Christ that man who is generated in Adam”(Letters 98:2)

Origen also wrote in the third century that 'according to the usage of Church, baptism is given even to infants'(Homilies on Leviticus, 8:3:11,AD 244).

The council of Carthage, in 253, condemned the opinion that baptism should be withheld from infants until the eight day after birth.

Later Augustine taught, 'The custom of Mother Church in baptising infants is certainly not to be scorned ....nor is to be believed that its tradition is anything except apostolic' (Literal Interpretation of Genesis 10:23:39, AD 408) - What is the Biblical basis for the Catholic teaching that infant baptism cleanses from original sin?

Original sin in Scripture

The classical text is Romans 5:12 sqq. In the preceding part the apostle treats of justification by Jesus Christ, and to put in evidence the fact of His being the one Saviour, he contrasts with this Divine Head of mankind the human head who caused its ruin. The question of original sin, therefore, comes in only incidentally. St. Paul supposes the idea that the faithful have of it from his oral instructions, and he speaks of it to make them understand the work of Redemption. This explains the brevity of the development and the obscurity of some verses. Original Sin

Original sin in tradition

On account of a superficial resemblance between the doctrine of original sin and the Manichaean theory of our nature being evil, the Pelagians accused the Catholics and St. Augustine of Manichaeism. For the accusation and its answer see "Contra duas epist. Pelag.", I, II, 4; V, 10; III, IX, 25; IV, III. In our own times this charge has been reiterated by several critics and historians of dogma who have been influenced by the fact that before his conversion St. Augustine was a Manichaean. They do not identify Manichaeism with the doctrine of original sin, but they say that St. Augustine, with the remains of his former Manichaean prejudices, created the doctrine of original sin unknown before his time.

It is not true that the doctrine of original sin does not appear in the works of the pre-Augustinian Fathers. On the contrary, their testimony is found in special works on the subject. Nor can it be said, as Harnack maintains, that St. Augustine himself acknowledges the absence of this doctrine in the writings of the Fathers. St. Augustine invokes the testimony of eleven Fathers, Greek as well as Latin (Contra Jul., II, x, 33). Baseless also is the assertion that before St. Augustine this doctrine was unknown to the Jews and to the Christians; as we have already shown, it was taught by St. Paul. It is found in the fourth Book of Esdras, a work written by a Jew in the first century after Christ and widely read by the Christians. This book represents Adam as the author of the fall of the human race (vii, 48), as having transmitted to all his posterity the permanent infirmity, the malignity, the bad seed of sin (iii, 21, 22; iv, 30). Protestants themselves admit the doctrine of original sin in this book and others of the same period (see Sanday, "The International Critical Commentary: Romans", 134, 137; Hastings, "A Dictionary of the Bible", I, 841). It is therefore impossible to make St. Augustine, who is of a much later date, the inventor of original sin.

That this doctrine existed in Christian tradition before St. Augustine's time is shown by the practice of the Church in the baptism of children. The Pelagians held that baptism was given to children, not to remit their sin, but to make them better, to give them supernatural life, to make them adoptive sons of God, and heirs to the Kingdom of Heaven (see St. Augustine, Of Sin and Merit I.18). The Catholics answered by citing the Nicene Creed, "Confiteor unum baptisma in remissionem peccatorum". - Original Sin

It seems the Early Church probably practiced infant baptism from apostolic times due to concept of original sin as it was understood at the time.

Related Questions:

Has the term “born again” always been synonymous with baptism with the Christians of the Early Church?

What is the earliest explicit mention of infant baptism?