Objection 1. It would seem that one may be punished for another's sin. For it is written (Exodus 20:5): "I am... God... jealous, visiting the iniquity of the fathers upon the children, unto the third and fourth generation of them that hate Me"; and (Matthew 23:35): "That upon you may come all the just blood that hath been shed upon the earth."
Objection 2. Further, human justice springs from Divine justice. Now, according to human justice, children are sometimes punished for their parents, as in the case of high treason. Therefore also according to Divine justice, one is punished for another's sin.
Objection 3. Further, if it be replied that the son is punished, not for the father's sin, but for his own, inasmuch as he imitates his father's wickedness; this would not be said of the children rather than of outsiders, who are punished in like manner as those whose crimes they imitate. It seems, therefore, that children are punished, not for their own sins, but for those of their parents.
On the contrary, It is written (Ezekiel 18:20): "The son shall not bear the iniquity of the father."
I answer that, If we speak of that satisfactory punishment, which one takes upon oneself voluntarily, one may bear another's punishment, in so far as they are, in some way, one, as stated above (Article 7). If, however, we speak of punishment inflicted on account of sin, inasmuch as it is penal, then each one is punished for his own sin only, because the sinful act is something personal. But if we speak of a punishment that is medicinal, in this way it does happen that one is punished for another's sin. For it has been stated (7) that ills sustained in bodily goods or even in the body itself, are medicinal punishments intended for the health of the soul. Wherefore there is no reason why one should not have such like punishments inflicted on one for another's sin, either by God or by man; e.g. on children for their parents, or on servants for their masters, inasmuch as they are their property so to speak; in such a way, however, that, if the children or the servants take part in the sin, this penal ill has the character of punishment in regard to both the one punished and the one he is punished for. But if they do not take part in the sin, it has the character of punishment in regard to the one for whom the punishment is borne, while, in regard to the one who is punished, it is merely medicinal (except accidentally, if he consent to the other's sin), since it is intended for the good of his soul, if he bears it patiently.
With regard to spiritual punishments, these are not merely medicinal, because the good of the soul is not directed to a yet higher good. Consequently no one suffers loss in the goods of the soul without some fault of his own. Wherefore Augustine says (Ep. ad Avit.) [Ep. ad Auxilium, ccl.], such like punishments are not inflicted on one for another's sin, because, as regards the soul, the son is not the father's property. Hence the Lord assigns the reason for this by saying (Ezekiel 18:4): "All souls are Mine."
Reply to Objection 1. Both the passages quoted should, seemingly, be referred to temporal or bodily punishments, in so far as children are the property of their parents, and posterity, of their forefathers. Else, if they be referred to spiritual punishments, they must be understood in reference to the imitation of sin, wherefore in Exodus these words are added, "Of them that hate Me," and in the chapter quoted from Matthew (verse 32) we read: "Fill ye up then the measure of your fathers." The sins of the fathers are said to be punished in their children, because the latter are the more prone to sin through being brought up amid their parents' crimes, both by becoming accustomed to them, and by imitating their parents' example, conforming to their authority as it were. Moreover they deserve heavier punishment if, seeing the punishment of their parents, they fail to mend their ways. The text adds, "to the third and fourth generation," because men are wont to live long enough to see the third and fourth generation, so that both the children can witness their parents' sins so as to imitate them, and the parents can see their children's punishments so as to grieve for them.
Reply to Objection 2. The punishments which human justice inflicts on one for another's sin are bodily and temporal. They are also remedies or medicines against future sins, in order that either they who are punished, or others may be restrained from similar faults.
Reply to Objection 3. Those who are near of kin are said to be punished, rather than outsiders, for the sins of others, both because the punishment of kindred redounds somewhat upon those who sinned, as stated above, in so far as the child is the father's property, and because the examples and the punishments that occur in one's own household are more moving. Consequently when a man is brought up amid the sins of his parents, he is more eager to imitate them, and if he is not deterred by their punishments, he would seem to be the more obstinate, and, therefore, to deserve more severe punishment.
Source: Summa Theologica (New Advent)