Spent an evening in Bari decade and a half ago (coming in from Dubrovnik by ferry before heading onward the next morning). The (rather compact) old town by the port is the attraction (not that we got there) but the "new" town seemed rather nice and lively. Found a pizzeria serving assortments of delicious hot/warm antipastos which seem to be a rarity. Brindisi (flew out of there after the end of a cruise ~50 years ago) seems to be more of a port city than Bari (which is also a port city) but was also the terminus of the Via Appia.

Bari airport is a bit further out of town than Brindisi (we bussed to pick up the rental car at the former) while the latter is pretty much right in town.

You could get a car to explore Matera (the old town is excavated from tufa) and Alberobello (trullis) and stay out there. If you are arriving early enough in the day, I'd just rent a car (driving there isn't that bad) and stay in either of those two places.