There's nothing to climb in the Badlands. They're mostly like crumbly mud. You may be able to pick your way up some of the sides, but they're really not like rock. There are some really cool hiking trails, though, and some, like the Notch trail and the Saddle Pass trail are kind of challenging. It's usually a bit hot, and there's very little shade, so dress accordingly, and carry plenty of water. Also, if you have a dog, be aware that they aren't allowed on (or off) any trails, and must stay within 30 feet of the road or parking lot at all times.
I travel through the Badlands several times a year while going between Wisconsin and the Black Hills. I used to hike the trails pretty regularly, but since the dog rules have changed, and since I won't leave my dog in the car while I hike, I'm pretty much confined to roadways. If you hike there, be careful of hollow spaces under the rocks. Since I started going there in '72, a lot of these holes have gotten really big, and some extend under the trails, and they're hard to see because of the way the rocks crumble into them.
But, in answer to your question, no, there's nothing in the Badlands that's good to climb.

I didn't mean to change your mind about stopping there, The Badlands Park is one of my favorite places. Even if you can't climb, there's a lot of really interesting hiking. And if you're going east on I-90 anyway, you can detour into the Badlands at Wall and come out the east side while only adding an hour or two to your trip, and you'd never regret seeing the views. I know you've been there before, but it must have been a while ago, or you'd remember that they weren't good to climb.

By the way, if you're going east on I-90, I hope you're planning to stop at Devil's Tower. I've been to the top of it and done a lot of climbing there, and it's got some of the best climbing in the world.

Are you a college student? Just curious.