The Badlands get very hot in the Summer and there is no shade. If you stay overnight you will want a campground with at least electric hookups. Others have suggested all of the local campgrounds already, so I won't repeat.
IMO if you camp in the Black Hills it is a waste of time to drive back and forth for a day trip visit to the Badlands, especially since you will be coming from the east in the first place. The Badlands can be done in a 1/2 day drive through with some short hikes if it isn't too hot for that. Wall Drug is unique and is good for an hour or so. After the Badlands you will appreciate the free ice water.