Objection 1. It would seem that it is not fitting to say that when Christ was baptized the Holy Ghost came down on Him in the form of a dove. For the Holy Ghost dwells in man by grace. But the fulness of grace was in the Man-Christ from the beginning of His conception, because He was the "Only-begotten of the Father," as is clear from what has been said above (7, 12; 34, 1). Therefore the Holy Ghost should not have been sent to Him at His baptism.
Objection 2. Further, Christ is said to have "descended" into the world in the mystery of Incarnation, when "He emptied Himself, taking the form of a servant" (Philippians 2:7). But the Holy Ghost did not become incarnate. Therefore it is unbecoming to say that the Holy Ghost "descended upon Him."
Objection 3. Further, that which is accomplished in our baptism should have been shown in Christ's baptism, as in an exemplar. But in our baptism no visible mission of the Holy Ghost takes place. Therefore neither should a visible mission of the Holy Ghost have taken place in Christ's baptism.
Objection 4. Further, the Holy Ghost is poured forth on others through Christ, according to John 1:16: "Of His fulness we all have received." But the Holy Ghost came down on the apostles in the form, not of a dove, but of fire. Therefore neither should He have come down on Christ in the form of a dove, but in the form of fire.
On the contrary, It is written (Luke 3:22): "The Holy Ghost descended in a bodily shape as a dove upon Him."
I answer that, What took place with respect to Christ in His baptism, as Chrysostom says (Hom. iv in Matth. [From the supposititious Opus Imperfectum), "is connected with the mystery accomplished in all who were to be baptized afterwards." Now, all those who are baptized with the baptism of Christ receive the Holy Ghost, unless they approach unworthily; according to Matthew 3:11: "He shall baptize you in the Holy Ghost." Therefore it was fitting that when our Lord was baptized the Holy Ghost should descend upon Him.
Reply to Objection 1. As Augustine says (De Trin. xv): "It is most absurd to say that Christ received the Holy Ghost, when He was already thirty years old: for when He came to be baptized, since He was without sin, therefore was He not without the Holy Ghost. For if it is written of John that 'he shall be filled with the Holy Ghost from his mother's womb,' what must we say of the Man-Christ, whose conception in the flesh was not carnal, but spiritual? Therefore now," i.e. at His baptism, "He deigned to foreshadow His body," i.e. the Church, "in which those who are baptized receive the Holy Ghost in a special manner."
Reply to Objection 2. As Augustine says (De Trin. ii), the Holy Ghost is said to have descended on Christ in a bodily shape, as a dove, not because the very substance of the Holy Ghost was seen, for He is invisible: nor as though that visible creature were assumed into the unity of the Divine Person; since it is not said that the Holy Ghost was the dove, as it is said that the Son of God is man by reason of the union. Nor, again, was the Holy Ghost seen under the form of a dove, after the manner in which John saw the slain Lamb in the Apocalypse (5:6): "For the latter vision took place in the spirit through spiritual images of bodies; whereas no one ever doubted that this dove was seen by the eyes of the body." Nor, again, did the Holy Ghost appear under the form of a dove in the sense in which it is said (1 Corinthians 10:4): "'Now, the rock was Christ': for the latter had already a created existence, and through the manner of its action was called by the name of Christ, whom it signified: whereas this dove came suddenly into existence, to fulfil the purpose of its signification, and afterwards ceased to exist, like the flame which appeared in the bush to Moses."
Hence the Holy Ghost is said to have descended upon Christ, not by reason of His being united to the dove: but either because the dove itself signified the Holy Ghost, inasmuch as it "descended" when it came upon Him; or, again, by reason of the spiritual grace, which is poured out by God, so as to descend, as it were, on the creature, according to James 1:17: "Every best gift and every perfect gift is from above, coming down from the Father of lights."
Reply to Objection 3. As Chrysostom says (Hom. xii in Matth.): "At the beginning of all spiritual transactions sensible visions appear, for the sake of them who cannot conceive at all an incorporeal nature... so that, though afterwards no such thing occur, they may shape their faith according to that which has occurred once for all." And therefore the Holy Ghost descended visibly, under a bodily shape, on Christ at His baptism, in order that we may believe Him to descend invisibly on all those who are baptized.
Reply to Objection 4. The Holy Ghost appeared over Christ at His baptism, under the form of a dove, for four reasons. First, on account of the disposition required in the one baptized--namely, that he approach in good faith: since! as it is written (Wisdom 1:5): "The holy spirit of discipline will flee from the deceitful." For the dove is an animal of a simple character, void of cunning and deceit: whence it is said (Matthew 10:16): "Be ye simple as doves."
Secondly, in order to designate the seven gifts of the Holy Ghost, which are signified by the properties of the dove. For the dove dwells beside the running stream, in order that, on perceiving the hawk, it may plunge in and escape. This refers to the gift of wisdom, whereby the saints dwell beside the running waters of Holy Scripture, in order to escape the assaults of the devil. Again, the dove prefers the more choice seeds. This refers to the gift of knowledge, whereby the saints make choice of sound doctrines, with which they nourish themselves. Further, the dove feeds the brood of other birds. This refers to the gift of counsel, with which the saints, by teaching and example, feed men who have been the brood, i.e. imitators, of the devil. Again, the dove tears not with its beak. This refers to the gift of understanding, wherewith the saints do not rend sound doctrines, as heretics do. Again, the dove has no gall. This refers to the gift of piety, by reason of which the saints are free from unreasonable anger. Again, the dove builds its nest in the cleft of a rock. This refers to the gift of fortitude, wherewith the saints build their nest, i.e. take refuge and hope, in the death wounds of Christ, who is the Rock of strength. Lastly, the dove has a plaintive song. This refers to the gift of fear, wherewith the saints delight in bewailing sins.
Thirdly, the Holy Ghost appeared under the form of a dove on account of the proper effect of baptism, which is the remission of sins and reconciliation with God: for the dove is a gentle creature. Wherefore, as Chrysostom says, (Hom. xii in Matth.), "at the Deluge this creature appeared bearing an olive branch, and publishing the tidings of the universal peace of the whole world: and now again the dove appears at the baptism, pointing to our Deliverer."
Fourthly, the Holy Ghost appeared over our Lord at His baptism in the form of a dove, in order to designate the common effect of baptism--namely, the building up of the unity of the Church. Hence it is written (Ephesians 5:25-27): "Christ delivered Himself up... that He might present... to Himself a glorious Church, not having spot or wrinkle, or any such thing... cleansing it by the laver of water in the word of life." Therefore it was fitting that the Holy Ghost should appear at the baptism under the form of a dove, which is a creature both loving and gregarious. Wherefore also it is said of the Church (Canticles 6:8): "One is my dove."
But on the apostles the Holy Ghost descended under the form of fire, for two reasons. First, to show with what fervor their hearts were to be moved, so as to preach Christ everywhere, though surrounded by opposition. And therefore He appeared as a fiery tongue. Hence Augustine says (Super Joan., Tract. vi): Our Lord "manifests" the Holy Ghost "visibly in two ways"--namely, "by the dove corning upon the Lord when He was baptized; by fire, coming upon the disciples when they were met together... In the former case simplicity is shown, in the latter fervor... We learn, then, from the dove, that those who are sanctified by the Spirit should be without guile: and from the fire, that their simplicity should not be left to wax cold. Nor let it disturb anyone that the tongues were cloven... in the dove recognize unity."
Secondly, because, as Chrysostom says (Gregory, Hom. xxx in Ev.): "Since sins had to be forgiven," which is effected in baptism, "meekness was required"; this is shown by the dove: "but when we have obtained grace we must look forward to be judged"; and this is signified by the fire.
Source: Summa Theologica (New Advent)