Objection 1. It seems that vengeance should not be wrought by means of punishments customary among men. For to put a man to death is to uproot him. But our Lord forbade (Matthew 13:29) the uprooting of the cockle, whereby the children of the wicked one are signified. Therefore sinners should not be put to death.
Objection 2. Further, all who sin mortally seem to be deserving of the same punishment. Therefore if some who sin mortally are punished with death, it seems that all such persons should be punished with death: and this is evidently false.
Objection 3. Further, to punish a man publicly for his sin seems to publish his sin: and this would seem to have a harmful effect on the multitude, since the example of sin is taken by them as an occasion for sin. Therefore it seems that the punishment of death should not be inflicted for a sin.
On the contrary, These punishments are fixed by the divine law as appears from what we have said above (I-II, 105, 2).
I answer that, Vengeance is lawful and virtuous so far as it tends to the prevention of evil. Now some who are not influenced by motive of virtue are prevented from committing sin, through fear of losing those things which they love more than those they obtain by sinning, else fear would be no restraint to sin. Consequently vengeance for sin should be taken by depriving a man of what he loves most. Now the things which man loves most are life, bodily safety, his own freedom, and external goods such as riches, his country and his good name. Wherefore, according to Augustine's reckoning (De Civ. Dei xxi), "Tully writes that the laws recognize eight kinds of punishment": namely, "death," whereby man is deprived of life; "stripes," "retaliation," or the loss of eye for eye, whereby man forfeits his bodily safety; "slavery," and "imprisonment," whereby he is deprived of freedom; "exile" whereby he is banished from his country; "fines," whereby he is mulcted in his riches; "ignominy," whereby he loses his good name.
Reply to Objection 1. Our Lord forbids the uprooting of the cockle, when there is fear lest the wheat be uprooted together with it. But sometimes the wicked can be uprooted by death, not only without danger, but even with great profit, to the good. Wherefore in such a case the punishment of death may be inflicted on sinners.
Reply to Objection 2. All who sin mortally are deserving of eternal death, as regards future retribution, which is in accordance with the truth of the divine judgment. But the punishments of this life are more of a medicinal character; wherefore the punishment of death is inflicted on those sins alone which conduce to the grave undoing of others.
Reply to Objection 3. The very fact that the punishment, whether of death or of any kind that is fearsome to man, is made known at the same time as the sin, makes man's will avers to sin: because the fear of punishment is greater than the enticement of the example of sin.
Source: Summa Theologica (New Advent)