Old Fort is a lovely gated community on the western end of the island, and about 10 minutes from the airport. There is 24 hour security, and depending on who the owner of the villa is, you may have access to Old Fort Club with its beach, restaurant and amenities. Most do not, but it does not hurt to ask if your renter will offer "guest privileges" for a fee.
It is well out of the resort area, and you will need to plan for transportation. I normally do not recommend renting cars, but you can easily rent a car, as parking will not be an issue at the rental, or at the Wyndham for short periods. There is free parking across the street, or you can valet for $10. Buses do run (#12) and the bus stop is outside the western gate of OFB in front of the gas station. The buses stop running after 5pm. Taxis are available, though you will need to arrange this with the security booth, as they cannot just come and pick you up without checking it at the gate first. It is a lovely, quiet and family oriented community, so will be relaxing.
The Wyndham is about 10-15 minutes from OFB. There is a very nice grocery store called Solomon's Fresh Market on the way from the airport and only minutes from OFB with a nice bakery, meat counter, organic choices and great exotic snacks. A bit more expensive, but worth it.