When Oswald confesses to his mom that he has syphilis, he echoes the doctor's biblical diagnosis that "The sins of the fathers are visited upon the children" (2.270). While you can't get syphilis from your father (unless you sleep with him), the symbol is what's important here. Syphilis does represent what Oswald inherited from his father, but it's not really "sin."
Oswald inherited his father's buoyant, joyful nature, a nature suffocated and defiled by the puritanical Norwegian society. In this play, syphilis is a symbol of what happens when an important, natural life force (i.e., sex) is driven out of "respectable homes" and into dark corners like the brothel Engstrand wants to start. Mrs. Alving is a main player in forcing her husband into a twisted expression of his passion. His infection comes back to haunt her, as she faces the decision of whether or not to kill her diseased son.