Fly to Bari and take the airport bus to Matera.
If the shuttle from the airport isn't convenient to your schedule, take the train from Bari. Matera isn't served by Trenitalia, but by a regional train.
Ferrovie Appulo Lucane S.r.l.:
Matera and its sassi are wonderful – A UNESCO site!!
Pollypocket, the closest airport to Matera is BARI, you can rent a car in Bari and drive to Matera it's 65kms far approx 1hr, no highways but regional roads.
If possible, take the bus or train to Matera and hire the car when you depart Matera - Parking is a challenge in the historic centre. Matera's historic centre has some very charming 'sassi' hotels - Don't miss the opportunity to sleep in a sassi!
It looks like you might be coming from the US so getting a flight into Bari could be tricky depending on the time of year. You can do it but might require switching carriers in Europe, or even airports.
Though Bari is the closest airport you may need to consider Naples for flights from the US. You can get a car in Naples, train connections would take too long, then drive over. It is about a 2.5 hour drive but some wonderful mountain/valley scenery.
If you fly to Rome the drive would be about 6 hours, or more with jet-lag.
I did this trip last June, flying into Rome and driving to Matera. By the time I got there I was nodding off. Do take jet lag into consideration. It sounds like you will have more than one driver though.
Vagabonda is right (as usual) about the sassi experience. I stayed at Locanda San Martino (http://www.locandadisanmartino.it/en/hotel/) and was extremely pleased. They even have a parking garage. Invest in insurance because it is almost unavoidable to get new scratches on the car.
I hope you are planning on spending more than one night there. The town is so interesting that you can't see it all in half a day. I stayed two nights and wished that I had at least one more day.
It’s not a problem switching carriers once you arrive in Europe. If you don’t book straight through to Bari on your transatlantic ticket, it wouldn’t be difficult finding flights to Bari – Alitalia, Air One, easyJet, Ryaniar, Meridiana….
Beware baggage allowances on low-cost airlines.
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