Rafter J is a very nice campground, a bit higher in price than some others, but very nicely laid out and with a pool and wifi. Very centrally located. Easy access off a main road but set far back away from traffic noise.
The Palmer Gulch KOA is the one previous posters are referring to, it's very close to Mt. Rushmore and centrally located, but it's located down a curvy mountain road (same road to Mt. Rushmore) so that will add some drive time every day. It's also priced a bit higher than other cg's in the area, but many activities for kids. The downside is it's a KOA and packed closely together, and super busy as KOA's are known to be.
We like to base camp in Custer as it's centrally located to our favorite things. We've stayed at two cg's there. Beaver Lake Campground is just west of town, very nice people working there. Some sites are close but many are spaced nicely. There is wifi and cable and a small pool, along with a waterslide (at an extra charge).
The other Custer cg we've stayed at is The Roost Resort, it's a very small cg but a beautiful location, spaced out nicely, and the owners are very nice people. It's in a great location of between the town of Custer and Custer State Park.