The oversized line up of a mountain carved into faces must have a mass appeal specifically for campers: almost every single campground in South Dakota is within a few miles of Mt. Rushmore.
It is not really a surprise that South Dakota's biggest tourist attraction would draw many campgrounds to the area, Mt. Rushmore brings a large number of tourists from all over the world through the state each year. Besides, most of South Dakota's most interesting sites are clumped into the southwest corner of the state, and so, so are all the tourists. Only the campground at Sheridan Lake is any distance away from the rest of the campsites.
In addition to Mt. Rushmore, campers can spend time in the famous Badlands National Park. Badlands Park is famous for its fossil beds and for its extremely rugged conditions. The fossils lend science a glimpse of the evolutionary patterns of some of the many mammals and plants that occupied the great plains of North America and can be viewed by the public through one of the ranger-guided hikes.
Badlands is a rare park in the United States in that it still has an open backcountry policy throughout the park. Hikers and campers can roam uninhibited through the entire Badlands area and set up camp where ever they feel is the best spot. Conditions in Badlands can be severe, however, with temperatures ranging from -40 F to 116 F depending on the season and the weather system moving through. It is recommended that travelers check in at the visitors center before their trip to receive some basic safety information and learn a little bit about the terrain. Other than that, you're free to roam and wander as you please in between camping.
Custer State Park is also in the same vicinity as most of the campgrounds and the park accommodates most any activity you would want to try. Archery, off-roading, geocaching, disc golf and fishing are all options at the massive Custer State Park.
Most of the campgrounds in the southwestern part of the state also have space and hook ups for RV campers, who make up a large percentage of those visiting Mt. Rushmore and the surrounding parks and attractions. And if those are all booked in busy summer months, consider staying in South Dakota Motels.
South Dakota has many more National and State Parks that you can find more information on here: