If you are flying...
The two closest main airports are Jacksonville, FL Airport (JAX) and Savannah/Hilton Head Airport (SAV).
Savannah Airport is 80 miles north and is about 1h20mins drive on I-95 interstate. Jacksonville Airport is 70 miles south and is about 1h10mins drive on I-95 interstate.
If you need to connect anyway and want to fly in directly, Delta has 3 flights a day into Brunswick-Golden Isles Airport (BQK) from Atlanta. Brunswick Airport is 15 miles away and take 20-25 mins.
If you are flying private, McKinnon Saint Simons Island Airport is the closest general aviation airport, which is 4 miles from the Cloisters.