Objection 1. It seems that it is part of a deacon's duty to baptize. Because the duties of preaching and of baptizing were enjoined by our Lord at the same time, according to Matthew 28:19: "Going... teach ye all nations, baptizing them," etc. But it is part of a deacon's duty to preach the gospel. Therefore it seems that it is also part of a deacon's duty to baptize.
Objection 2. Further, according to Dionysius (Eccl. Hier. v) to "cleanse" is part of the deacon's duty. But cleansing from sins is effected specially by Baptism, according to Ephesians 5:26: "Cleansing it by the laver of water in the word of life." Therefore it seems that it belongs to a deacon to baptize.
Objection 3. Further, it is told of Blessed Laurence, who was a deacon, that he baptized many. Therefore it seems that it belongs to deacons to baptize.
On the contrary, Pope Gelasius I says (the passage is to be found in the Decrees, dist. 93): "We order the deacons to keep within their own province"; and further on: "Without bishop or priest they must not dare to baptize, except in cases of extreme urgency, when the aforesaid are a long way off."
I answer that, Just as the properties and duties of the heavenly orders are gathered from their names, as Dionysius says (Coel. Hier. vi), so can we gather, from the names of the ecclesiastical orders, what belongs to each order. Now "deacons" are so called from being "ministers"; because, to wit, it is not in the deacon's province to be the chief and official celebrant in conferring a sacrament, but to minister to others, his elders, in the sacramental dispensations. And so it does not belong to a deacon to confer the sacrament of Baptism officially as it were; but to assist and serve his elders in the bestowal of this and other sacraments. Hence Isidore says (Epist. ad Ludifred.): "It is a deacon's duty to assist and serve the priests, in all the rites of Christ's sacraments, viz. those of Baptism, of the Chrism, of the Paten and Chalice."
Reply to Objection 1. It is the deacon's duty to read the Gospel in church, and to preach it as one catechizing; hence Dionysius says (Eccl. Hier. v) that a deacon's office involves power over the unclean among whom he includes the catechumens. But to teach, i.e. to expound the Gospel, is the proper office of a bishop, whose action is "to perfect," as Dionysius teaches (Eccl. Hier. v); and "to perfect" is the same as "to teach." Consequently, it does not follow that the office of baptizing belongs to deacons.
Reply to Objection 2. As Dionysius says (Eccl. Hier. ii), Baptism has a power not only of "cleansing" but also of "enlightening." Consequently, it is outside the province of the deacon whose duty it is to cleanse only: viz. either by driving away the unclean, or by preparing them for the reception of a sacrament.
Reply to Objection 3. Because Baptism is a necessary sacrament, deacons are allowed to baptize in cases of urgency when their elders are not at hand; as appears from the authority of Gelasius quoted above. And it was thus that Blessed Laurence, being but a deacon, baptized.
Source: Summa Theologica (New Advent)