Objection 1. It seems that a woman cannot baptize. For we read in the acts of the Council of Carthage (iv): "However learned and holy a woman may be, she must not presume to teach men in the church, or to baptize." But in no case is a woman allowed to teach in church, according to 1 Corinthians 14:35: "It is a shame for a woman to speak in the church." Therefore it seems that neither is a woman in any circumstances permitted to baptize.
Objection 2. Further, to baptize belongs to those having authority. wherefore baptism should be conferred by priests having charge of souls. But women are not qualified for this; according to 1 Timothy 2:12: "I suffer not a woman to teach, nor to use authority over man, but to be subject to him [Vulgate: 'but to be in silence']." Therefore a woman cannot baptize.
Objection 3. Further, in the spiritual regeneration water seems to hold the place of the mother's womb, as Augustine says on John 3:4, "Can" a man "enter a second time into his mother's womb, and be born again?" While he who baptizes seems to hold rather the position of father. But this is unfitting for a woman. Therefore a woman cannot baptize.
On the contrary, Pope Urban II says (Decreta xxx): "In reply to the questions asked by your beatitude, we consider that the following answer should be given: that the baptism is valid when, in cases of necessity, a woman baptizes a child in the name of the Trinity."
I answer that, Christ is the chief Baptizer, according to John 1:33: "He upon Whom thou shalt see the Spirit descending and remaining upon Him, He it is that baptizeth." For it is written in Colossians 3 (cf. Galatians 3:28), that in Christ there is neither male nor female. Consequently, just as a layman can baptize, as Christ's minister, so can a woman.
But since "the head of the woman is the man," and "the head of... man, is Christ" (1 Corinthians 11:3), a woman should not baptize if a man be available for the purpose; just as neither should a layman in the presence of a cleric, nor a cleric in the presence of a priest. The last, however, can baptize in the presence of a bishop, because it is part of the priestly office.
Reply to Objection 1. Just as a woman is not suffered to teach in public, but is allowed to instruct and admonish privately; so she is not permitted to baptize publicly and solemnly, and yet she can baptize in a case of urgency.
Reply to Objection 2. When Baptism is celebrated solemnly and with due form, it should be conferred by a priest having charge of souls, or by one representing him. But this is not required in cases of urgency, when a woman may baptize.
Reply to Objection 3. In carnal generation male and female co-operate according to the power of their proper nature; wherefore the female cannot be the active, but only the passive, principle of generation. But in spiritual generation they do not act, either of them, by their proper power, but only instrumentally by the power of Christ. Consequently, on the same grounds either man or woman can baptize in a case of urgency.
If, however, a woman were to baptize without any urgency for so doing. there would be no need of rebaptism: as we have said in regard to laymen (3, ad 1). But the baptizer herself would sin, as also those who took part with her therein, either by receiving Baptism from her, or by bringing someone to her to be baptized.
Source: Summa Theologica (New Advent)