Objection 1. It seems that a layman cannot baptize. Because, as stated above (Article 2), to baptize belongs properly to the priestly order. But those things which belong to an order cannot be entrusted to one that is not ordained. Therefore it seems that a layman, who has no orders, cannot baptize.
Objection 2. Further, it is a greater thing to baptize, than to perform the other sacramental rites of Baptism, such as to catechize, to exorcize, and to bless the baptismal water. But these things cannot be done by laymen, but only by priests. Therefore it seems that much less can laymen baptize.
Objection 3. Further, just as Baptism is a necessary sacrament, so is Penance. But a layman cannot absolve in the tribunal of Penance. Neither, therefore, can he baptize.
On the contrary, Pope Gelasius I and Isidore say that "it is often permissible for Christian laymen to baptize, in cases of urgent necessity."
I answer that, It is due to the mercy of Him "Who will have all men to be saved" (1 Timothy 2:4) that in those things which are necessary for salvation, man can easily find the remedy. Now the most necessary among all the sacraments is Baptism, which is man's regeneration unto spiritual life: since for children there is no substitute, while adults cannot otherwise than by Baptism receive a full remission both of guilt and of its punishment. Consequently, lest man should have to go without so necessary a remedy, it was ordained, both that the matter of Baptism should be something common that is easily obtainable by all, i.e. water; and that the minister of Baptism should be anyone, even not in orders, lest from lack of being baptized, man should suffer loss of his salvation.
Reply to Objection 1. To baptize belongs to the priestly order by reason of a certain appropriateness and solemnity; but this is not essential to the sacrament. Consequently, if a layman were to baptize even outside a case of urgency; he would sin, yet he would confer the sacrament; nor would the person thus baptized have to be baptized again.
Reply to Objection 2. These sacramental rites of Baptism belong to the solemnity of, and are not essential to, Baptism. And therefore they neither should nor can be done by a layman, but only by a priest, whose office it is to baptize solemnly.
Reply to Objection 3. As stated above (65, 3,4), Penance is not so necessary as Baptism; since contrition can supply the defect of the priestly absolution which does not free from the whole punishment, nor again is it given to children. Therefore the comparison with Baptism does not stand, because its effect cannot be supplied by anything else.
Source: Summa Theologica (New Advent)