My vote would be Custer County Aiport (KCUT), located just S. of the town of Custer. I don't know if camping is allowed on the field, but there are numerous motels and campgrounds in the area. It's an easy rental car drive from Custer to all of the attractions (Crazy Horse Monument, Mt Rushmore, Hill City, etc.) and camping, hiking, fishing, etc. are close by at Custer State Park.
NOTE: Custer State Park has it's own airport (really just an airstrip), but camping is only permitted in the campgrounds.
See
http://www.airnav.com/airport/KCUT
As with any mountain flying, density altitude and crosswinds can be an issue, so plan a good alternate-have fun,
Mike