While it is true that the very first Christians to be baptized were adults, this is because adults were the first able to hear Christ’s message and act upon it. Often enough, those adults who were baptized by the apostles brought their families with them into the faith (cf. Acts 16:15, 16:33; 1 Cor. 1:16), an implication that baptism was not limited by the apostles to adults. It is also true enough that some early converts, such as Constantine the Great, the Roman emperor who legalized Christianity, waited until they were on their deathbeds to be baptized because baptism is a one-time only sacrament that washes away not just sin but all punishment due sin.
This was not encouraged by the Church, however. In fact, as the Church recognized the importance of baptism even for babies, one of the early Christian arguments was about how early babies could be baptized. Could they be baptized immediately after birth or did they need to wait until the eighth day, as was prescribed for circumcision under the old covenant? The Church determined that the eighth-day commandment for circumcision was not necessary for baptism and that babies could be baptized at any time after birth.
To this day, the Church still strongly encourages Catholic parents to have their children baptized as babies:
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 (CCC 1252).
As regards children who have died without baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them. Indeed, the great mercy of God who desires that all men should be saved, and Jesus’ tenderness toward children which caused him to say: “Let the children come to me, do not hinder them,” allow us to hope that there is a way of salvation for children who have died without baptism. All the more urgent is the Church’s call not to prevent little children coming to Christ through the gift of holy baptism (CCC 1261).
Early Teachings on Infant Baptism**