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
“[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