Well, for one, the baptizing of infants wasn't something that came after the doctrine of Original Sin was articulated and set forth by Augustine. If it was, then the East wouldn't also baptize infants, however they do just as we do in the West. The Augustinian Doctrine of Original Sin is unknown in the theology of the East because St. Augustine was never influential outside of the West.
Further, the baptizing of infants is well known and was practiced almost universally centuries before Augustine. We have in the writings of St. Hippolytus the standard practices of Christian Baptism, and concerning infants and children he writes:
"Baptize first the children, and if they can speak for themselves let them do so. Otherwise, let their parents or other relatives speak for them" - The Apostolic Tradition 21:4
The earliest opponent of infant baptism is Tertullian of Carthage, not because he does not believe baptism is not efficacious for infants and small children, but because Tertullian's theology was highly legalistic, believing it nearly impossible to find forgiveness after Baptism, thus he believes it better to delay Baptism as long as possible:
"According to circumstance and disposition and even age of the individual person, it may be better to delay Baptism; and especially so in the case of little children. Why, indeed, is it necessary -- if it be not a case of necessity -- that the sponsors to be thrust into danger, when they themselves may fail to fulfill their promises by reason of death, or when they may be disappointed by the growth of an evil disposition? Indeed the Lord says, 'Do not forbid them to come to me'" - On Baptism 18:4
All who write on the subject, even Tertullian who opposes it on his own idiosyncratic grounds, take the baptism of infants as a given, something widespread and having been done for quite some time.
Earlier than both Hippolytus and Tertullian, Irenaeus writes in his tome, Against Heresies,
"He [Jesus] came to save all through himself; all, I say, who through him are reborn in God: infants, and children, and youths, and old men. Therefore he passed through every age, becoming an infant for infants, sanctifying infants; a child for children, sanctifying those who are of that age . . . [so that] he might be the perfect teacher in all things, perfect not only in respect to the setting forth of truth, perfect also in respect to relative age" - Against Heresies 2:22:4
Infants, therefore, are to be included in the Church, as full members of Christ's Body; and the entryway is new birth in Christ, which happens in Holy Baptism.
So even if we remove Augustine entirely from the equation, and the doctrine of Original Sin as he defined it and put forth and which the West has embraced; even without Augustine and "Original Sin", the baptism of infants had long been an integral part of Christian practice, because infants and children were never to be outside of the purview of the Church's ministry, Christ's commission and command to make disciples of all nations excludes no one on the basis of age.
Christ does not discriminate, thus neither should we. Indeed, when Christ's own disciples sought to discriminate against the little ones, our Lord specifically aid, "Do not prevent the little ones from coming to Me, for to such as these belongs the kingdom". If Christ has spoken thus, why should we prevent the little ones from being with Christ?
Why deprive infants of Jesus Christ?
-CryptoLutheran