I spent 3 days in Brindisi 37 years ago (I had to do a major repair to my car)! The only "exciting" thing was to watch the cigarette smugglers preparing their boats, i.e., loading bottles of water, etc., but I never saw them coming back (during the night). I doubt, though, that the spectacle is still available; therefore, the only thing that is a "must see" is the scalinata at the very end of the Via Appia, at the waterfront. On the other hand, it is much closer to Lecce (not far from the tip of the heel of Italy), which is a very, very beautiful city, clearly worth a visit!