Q - I was wondering if you could tell me if there is reference in Scripture to baptizing babies. Do we baptize babies because of tradition or are we following Scripture?
A - Thanks for the question. There is no contradiction between Sacred Scripture and Sacred Tradition, thus the teaching is from both and ultimately, from Jesus. First we start with Scripture. In several places in the New Testament it speaks of baptizing families. One may assume that infants were a part of these families.
"After she and her household had been baptized, she offered us an invitation, "If you consider me a believer in the Lord, come and stay at my home," and she prevailed on us." - Acts 16:15
"He took them in at that hour of the night and bathed their wounds; then he and all his family were baptized at once." - Acts 16:33
"I did baptize also the household of Stephanas" - 1 Cor 1:16
But, alone, this does not "prove" infant baptism. We also see that all are called to be baptized and there is no age restrictions. On the day of Pentecost we see the following after Peter preaches:
Now when they heard this, they were cut to the heart, and they asked Peter and the other apostles, "What are we to do, my brothers?" 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 will receive the gift of the holy Spirit. For the promise is made to you and to your children and to all those far off, whomever the Lord our God will call." - Acts 2: 37-39Notice that Christians are called to baptism and that the promise of the Holy Spirit given in baptism is for "you and to your children".
We can also see a fulfillment of the Jewish practice of circumcision from the old covenant in the new covenant practice of baptism. Paul makes this comparison explicit:
"In him you were also circumcised with a circumcision not administered by hand, by stripping off the carnal body, with the circumcision of Christ. You were buried with him in baptism, in which you were also raised with him through faith in the power of God, who raised him from the dead." - Col 2: 11-12
So, we can see there is ample implicit Biblical evidence of infant baptism. Those that reject infant baptism are doing so on an argument from silence, that is they assume it is not valid, because it is not explicitly ordered in the Bible. Well, we also see that the early Christians practiced it and that they understood it to be perfectly consonant with Scripture. Thus, early Christians wrote:
"And when a child has been born to one of them[ie Christians], they give thanks to God[ie baptism]; and if moreover it happen to die in childhood, they give thanks to God the more, as for one who as passed through the world without sins." - Aristides,Apology,15(A.D. 140)
"And they shall baptise 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 (A.D. 215)
"‘And [Naaman] dipped himself . . . seven times in the Jordan’ [2 Kgs. 5:14]. It was not for nothing that Naaman of old, when suffering from leprosy, was purified upon his being baptized, but [this served] as an indication to us. For as we are lepers in sin, we are made clean, by means of the sacred water and the invocation of the Lord, from our old transgressions, being spiritually regenerated as newborn babes, even as the Lord has declared: ‘Except a man be born again through water and the Spirit, he shall not enter into the kingdom of heaven’ -John 3:5" - Iranaeus, Fragment (A.D. 190).
There are many more references here. I hope this helps.