Objection 1. It seems that to baptize is not part of the priestly office, but proper to that of bishops. Because, as stated above (1, Objection 1), the duties of teaching and baptizing are enjoined in the same precept (Matthew 28:19). But to teach, which is "to perfect," belongs to the office of bishop, as Dionysius declares (Eccl. Hier. v, vi). Therefore to baptize also belongs to the episcopal office.
Objection 2. Further, by Baptism a man is admitted to the body of the Christian people: and to do this seems consistent with no other than the princely office. Now the bishops hold the position of princes in the Church, as the gloss observes on Luke 10:1: indeed, they even take the place of the apostles, of whom it is written (Psalm 44:17): "Thou shalt make them princes over all the earth." Therefore it seems that to baptize belongs exclusively to the office of bishops.
Objection 3. Further, Isidore says (Epist. ad Ludifred.) that "it belongs to the bishop to consecrate churches, to anoint altars, to consecrate [conficere] the chrism; he it is that confers the ecclesiastical orders, and blesses the consecrated virgins." But the sacrament of Baptism is greater than all these. Therefore much more reason is there why to baptize should belong exclusively to the episcopal office.
On the contrary, Isidore says (De Officiis. ii): "It is certain that Baptism was entrusted to priests alone."
I answer that, Priests are consecrated for the purpose of celebrating the sacrament of Christ's Body, as stated above (Question 65, Article 3). Now that is the sacrament of ecclesiastical unity, according to the Apostle (1 Corinthians 10:17): "We, being many, are one bread, one body, all that partake of one bread and one chalice." Moreover, by Baptism a man becomes a participator in ecclesiastical unity, wherefore also he receives the right to approach our Lord's Table. Consequently, just as it belongs to a priest to consecrate the Eucharist, which is the principal purpose of the priesthood, so it is the proper office of a priest to baptize: since it seems to belong to one and the same, to produce the whole and to dispose the part in the whole.
Reply to Objection 1. Our Lord enjoined on the apostles, whose place is taken by the bishops, both duties, namely, of teaching and of baptizing, but in different ways. Because Christ committed to them the duty of teaching, that they might exercise it themselves as being the most important duty of all: wherefore the apostles themselves said (Acts 6:2): "It is not reason that we should leave the word of God and serve tables." On the other hand, He entrusted the apostles with the office of baptizing, to be exercised vicariously; wherefore the Apostle says (1 Corinthians 1:17): "Christ sent me not to baptize, but to preach the Gospel." And the reason for this was that the merit and wisdom of the minister have no bearing on the baptismal effect, as they have in teaching, as may be seen from what we have stated above (64, 1, ad 2; 5,9). A proof of this is found also in the fact that our Lord Himself did not baptize, but His disciples, as John relates (4:2). Nor does it follow from this that bishops cannot baptize; since what a lower power can do, that can also a higher power. Wherefore also the Apostle says (1 Corinthians 1:14-16) that he had baptized some.
Reply to Objection 2. In every commonwealth minor affairs are entrusted to lower officials, while greater affairs are restricted to higher officials; according to Exodus 18:22: "When any great matter soever shall fall out, let them refer it to thee, and let them judge the lesser matters only." Consequently it belongs to the lower officials of the state to decide matters concerning the lower orders; while to the highest it belongs to set in order those matters that regard the higher orders of the state. Now by Baptism a man attains only to the lowest rank among the Christian people: and consequently it belongs to the lesser officials of the Church to baptize, namely, the priests, who hold the place of the seventy-two disciples of Christ, as the gloss says in the passage quoted from Luke 10.
Reply to Objection 3. As stated above (Question 65, Article 3), the sacrament of Baptism holds the first place in the order of necessity; but in the order of perfection there are other greater sacraments which are reserved to bishops.
Source: Summa Theologica (New Advent)