The Badlands has its own campground now, but I've never stayed there, so I can't say anything about it.
Summers Carroll Campground is only about 10 miles away, and that's where most of the offroaders stay. The campground is nice and clean, has bathrooms and showers, and the owners are friendly. They have a little general store right in the office building, and POOL right behind it.
Address: 5509 North 200 East, Attica, IN 47918
Phone: (765) 762-2832
Website: http://www.summerscampground.com/
I stayed in Hotel Attica a couple times, and it was AWESOME! Its rooms are reasonably priced, and it's just down the street from the Badlands! The owners are super friendly and go out of their way to make you feel at home. The hotel has been remodeled about 2-3 years ago, it has a restaurant and a bar on the first floor, and, as far as I remember, the food is pretty good!
Address: 126 N Perry Street, Attica, IN 47914
Phone: (765) 762-3088
Website: http://www.hotelattica.com/
Another place I used to stay in is Boswell Motel.
Nothing really special about this place, just your basic motel. The good thing about it - it's located right on US41. It's a bit far from the Badlands, about 20 miles, but it's cheap.
Address: 307 S Old US Highway 41, Boswell, IN 47921
Phone: (765) 869-5060
They don't have a website.