Rapid City is the nearest big airport. Ellsworth Air Force
Base is a few miles away, and its easy to confuse the two: last year an airline
accidentally did just that and landed at Ellsworth. I guess they didn't have a
good moving map GPS.
If you want to fly over Mt. Rushmore and Crazy Horse, ask ATC for vectors and they'll steer you right past the monuments, especially on departure from KRAP. You can rent a car at Rapid City and drive to Mt. Rushmore: its less than an hour's drive.
There's a closer airport, called Custer County (CUT), but I've never landed there. I don't think you'd save a huge amount of time in terms of driving to the monument.
I was somewhat disappointed by Mt. Rushmore: they took a magnificent mountain range and carved it up for the sake of creating a tourist attraction. When you fly past those mountains you get a sense of how magnificent they truly are. Crazy Horse is an incredible endeavor that's appropriately named.
Enjoy your trip!