Brindisi Airport (BDS) is just 35 minutes away by car (38 kilometers) and Bari Airport (BRI) is 70 minutes away by car (100 kilometers).  I recommend checking fares for both because you can easily explore the Valle d’Itria from both airports.  If you need to rent a car, check both Bari and Brindisi for the best rates, but know that the one-way drop off fees tend to be high.