As @alexjo wrote in the comment, the two sentences are both correct, but have different meanings.
(1) "Non posso farlo. Infangherei la sua reputazione" means that I would sully his reputation.
(2) "Non posso farlo. Infangherebbe la sua reputazione" means that the action would sully his reputation, perhaps as an unintended but inevitable consequence.
In response to your final point, notice, however, that the clause "Se lo facessi" sound a bit awkward in:
(2)' "No posso farlo. Se lo facessi infangherebbe la sua reputazione"
While not exactly incorrect, the change of grammatical subject in the second sentence may even suggest that yet another subject has entered the picture. Perhaps someone else would take my action as an opportunity to sully his reputation.