Objection 1. It would seem that God became incarnate as a remedy for actual sins rather than for original sin. For the more grievous the sin, the more it runs counter to man's salvation, for which God became incarnate. But actual sin is more grievous than original sin; for the lightest punishment is due to original sin, as Augustine says (Contra Julian. v, 11). Therefore Incarnation of Christ is chiefly directed to taking away actual sins.
Objection 2. Further, pain of sense is not due to original sin, but merely pain of loss, as has been shown (I-II, 87, 5). But Christ came to suffer the pain of sense on the Cross in satisfaction for sins--and not the pain of loss, for He had no defect of either the beatific vision or fruition. Therefore He came in order to take away actual sin rather than original sin.
Objection 3. Further, as Chrysostom says (De Compunctione Cordis ii, 3): "This must be the mind of the faithful servant, to account the benefits of his Lord, which have been bestowed on all alike, as though they were bestowed on himself alone. For as if speaking of himself alone, Paul writes to the Galatians (2:20): 'Christ... loved me and delivered Himself for me.'" But our individual sins are actual sins; for original sin is the common sin. Therefore we ought to have this conviction, so as to believe that He has come chiefly for actual sins.
On the contrary, It is written (John 1:29): "Behold the Lamb of God, behold Him Who taketh away the sins [Vulgate: 'sin'] of the world."
I answer that, It is certain that Christ came into this world not only to take away that sin which is handed on originally to posterity, but also in order to take away all sins subsequently added to it; not that all are taken away (and this is from men's fault, inasmuch as they do not adhere to Christ, according to John 3:19: "The light is come into the world, and men loved darkness rather than the light"), but because He offered what was sufficient for blotting out all sins. Hence it is written (Romans 5:15-16): "But not as the offense, so also the gift... For judgment indeed was by one unto condemnation, but grace is of many offenses unto justification."
Moreover, the more grievous the sin, the more particularly did Christ come to blot it out. But "greater" is said in two ways: in one way "intensively," as a more intense whiteness is said to be greater, and in this way actual sin is greater than original sin; for it has more of the nature of voluntary, as has been shown (I-II, 81, 1). In another way a thing is said to be greater "extensively," as whiteness on a greater superficies is said to be greater; and in this way original sin, whereby the whole human race is infected, is greater than any actual sin, which is proper to one person. And in this respect Christ came principally to take away original sin, inasmuch as "the good of the race is a more Divine thing than the good of an individual," as is said Ethic. i, 2.
Reply to Objection 1. This reason looks to the intensive greatness of sin.
Reply to Objection 2. In the future award the pain of sense will not be meted out to original sin. Yet the penalties, such as hunger, thirst, death, and the like, which we suffer sensibly in this life flow from original sin. And hence Christ, in order to satisfy fully for original sin, wished to suffer sensible pain, that He might consume death and the like in Himself.
Reply to Objection 3. Chrysostom says (De Compunctione Cordis ii, 6): "The Apostle used these words, not as if wishing to diminish Christ's gifts, ample as they are, and spreading throughout the whole world, but that he might account himself alone the occasion of them. For what does it matter that they are given to others, if what are given to you are as complete and perfect as if none of them were given to another than yourself?" And hence, although a man ought to account Christ's gifts as given to himself, yet he ought not to consider them not to be given to others. And thus we do not exclude that He came to wipe away the sin of the whole nature rather than the sin of one person. But the sin of the nature is as perfectly healed in each one as if it were healed in him alone. Hence, on account of the union of charity, what is vouchsafed to all ought to be accounted his own by each one.
Source: Summa Theologica (New Advent)