The historic centre is small enough to cover on foot in an afternoon. The waterfront, the Roman column, and the main churches are all within a few minutes of each other. For the ferry terminal and the train station, the distances are still walkable but a taxi or local bus saves time if you are carrying luggage.
Local buses run by STP cover the wider city and some surrounding towns, though the service can be infrequent. Taxis are easy to find near the ferry terminal and the station. Renting a car is worthwhile if you plan to use Brindisi as a base for exploring the Salento: villages like Ostuni and Alberobello to the north, or the Adriatic coast to the south, are all more rewarding with your own transport.