Objection 1. It seems that not all are bound to receive Baptism. For Christ did not narrow man's road to salvation. But before Christ's coming men could be saved without Baptism: therefore also after Christ's coming.
Objection 2. Further, Baptism seems to have been instituted principally as a remedy for original sin. Now, since a man who is baptized is without original sin, it seems that he cannot transmit it to his children. Therefore it seems that the children of those who have been baptized, should not themselves be baptized.
Objection 3. Further, Baptism is given in order that a man may, through grace, be cleansed from sin. But those who are sanctified in the womb, obtain this without Baptism. Therefore they are not bound to receive Baptism.
On the contrary, It is written (John 3:5): "Unless a man be born again of water and the Holy Ghost, he cannot enter into the kingdom of God." Again it is stated in De Eccl. Dogm. xli, that "we believe the way of salvation to be open to those only who are baptized."
I answer that, Men are bound to that without which they cannot obtain salvation. Now it is manifest that no one can obtain salvation but through Christ; wherefore the Apostle says (Romans 5:18): "As by the offense of one unto all men unto condemnation; so also by the justice of one, unto all men unto justification of life." But for this end is Baptism conferred on a man, that being regenerated thereby, he may be incorporated in Christ, by becoming His member: wherefore it is written (Galatians 3:27): "As many of you as have been baptized in Christ, have put on Christ." Consequently it is manifest that all are bound to be baptized: and that without Baptism there is no salvation for men.
Reply to Objection 1. At no time, not even before the coming of Christ, could men be saved unless they became members of Christ: because, as it is written (Acts 4:12), "there is no other name under heaven given to men, whereby we must be saved." But before Christ's coming, men were incorporated in Christ by faith in His future coming: of which faith circumcision was the "seal," as the Apostle calls it (Romans 4:11): whereas before circumcision was instituted, men were incorporated in Christ by "faith alone," as Gregory says (Moral. iv), together with the offering of sacrifices, by means of which the Fathers of old made profession of their faith. Again, since Christ's coming, men are incorporated in Christ by faith; according to Ephesians 3:17: "That Christ may dwell by faith in your hearts." But faith in a thing already present is manifested by a sign different from that by which it was manifested when that thing was yet in the future: just as we use other parts of the verb, to signify the present, the past, and the future. Consequently although the sacrament itself of Baptism was not always necessary for salvation, yet faith, of which Baptism is the sacrament, was always necessary.
Reply to Objection 2. As we have stated in I-II, 81, 3, ad 2, those who are baptized are renewed in spirit by Baptism, while their body remains subject to the oldness of sin, according to Romans 8:10: "The body, indeed, is dead because of sin, but the spirit liveth because of justification." Wherefore Augustine (Contra Julian. vi) proves that "not everything that is in man is baptized." Now it is manifest that in carnal generation man does not beget in respect of his soul, but in respect of his body. Consequently the children of those who are baptized are born with original sin; wherefore they need to be baptized.
Reply to Objection 3. Those who are sanctified in the womb, receive indeed grace which cleanses them from original sin, but they do not therefore receive the character, by which they are conformed to Christ. Consequently, if any were to be sanctified in the womb now, they would need to be baptized, in order to be conformed to Christ's other members by receiving the character.
Source: Summa Theologica (New Advent)