The nearest bank is a little under half a mile from the airport (RBTT - on the left shortly after the Coco Reef Resort). However, there are two ATM machines at the airport. See menu: Infomation > Money Matters > ATM for details.
Steve Wooler myTobago.info - the definitive Visitor Guide to Tobago
There is actually a branch of Republic Bank in the airport compound - in the same building as the bar/cafeteria, opposite intenational departures. There is also a bureau de change in the same building (next to Tourist Information office)
Sorry Steve, have to beg to differ on that one. There is actually a bank at the airport as we used it a couple of times last year to change up travellers cheques and withdraw cash. Its within the airport compound (just across the road from arrivals) and there is also a very nice gift and souvenir next door. Rosyv