Great question – here is the answer . . .
Scripture tells us that the ENTIRE households of Cornelius, Stephanas and the Philippian Jailer (Acts 10, Acts 16, and 1 Corinthians 1) were Baptized.
Entire households contained people of ALL ages – young, middle-aged, old – and, yes, even BABIES.
It was very common in first century culture to have multi-generational households. That is to say, parents, children, grandparents, etc.
The UNANIMOUS consensus of the Early Church wirings is that the Baptism of infants was an APOSTOLIC teaching that had been handed down to them.
Irenaeus
"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 [A.D. 189]).
Hippolytus
"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 21:16 [A.D. 215]).
Origen
"Every soul that is born into flesh is soiled by the filth of wickedness and sin. . . . In the Church, baptism is given for the remission of sins, and, according to the usage of the Church, baptism is given even to infants. If there were nothing in infants which required the remission of sins and nothing in them pertinent to forgiveness, the grace of baptism would seem superfluous" (Homilies on Leviticus 8:3 [A.D. 248]).
"The Church received from the apostles the tradition of giving baptism even to infants. The apostles, to whom were committed the secrets of the divine sacraments, knew there are in everyone innate strains of [original] sin, which must be washed away through water and the Spirit" (Commentaries on Romans 5:9 [A.D. 248]).
Gregory of Nazianz
"Do you have an infant child? Allow sin no opportunity; rather, let the infant be sanctified from childhood.
Augustine
"The custom of Mother Church in baptizing infants is certainly not to be scorned, nor is it to be regarded in any way as superfluous, nor is it to be believed that its tradition is anything except apostolic" (The Literal Interpretation of Genesis 10:23:39 [A.D. 408]).