Hi Cheryl,
A car is pretty much needed to get around down here. Trains and buses aren't frequent, some places not served at all. (For example, Matera is the only provincial capital in Italy without rail service!)
I like the Valle d'Itria area of Puglia myself. The towns are lovely, there are things to see, good food, and the Adriatic is close by. Martina Franca or Locorotondo as a base, in my opinion. Polignano a Mare is a great seaside town to visit.
With only 3 or 4 days, it might be hard to see much of Basilicata from there apart from Matera, unless they split it up, depending on where they depart from. If they go back to Rome or Naples, a night in central Basilicata could work, stopping at Matera, then spend the night near Castelmezzano, which is truly beautiful. But depends on their travel style and overall itinerary.