If the paint has bubbled away from the gypsum, there's a good chance the paper has too. If you have latex paint and the leak is stopped, it'll slowly dry but you'll be left with loose areas that might need scraping followed with a thin coat of mud and re-painting. If it's oil-based paint (don't use oil-based paint in a basement...) you probably need to scrape that section off to even allow it to dry. Once it is, asses the damage and repair/replace as needed. If you do replace it, go for drywall that's intended for moist areas. It's not intended to stay wet, but does hold up better if it happens to get a little water in it.

Thanks for the advice; it's been a while since I posted. Turned out it was a little worse than initially thought. I found out that water had soaked into the carpet and underpad. I pulled back the carpet and underpad until I saw dry concrete. Then I ran fans and a dehumidifier for a few days until everything dried out.

Now I am wondering what steps I should take before putting the carpet back. Since the water was sitting there for about a week, I am worried about mold or other damage. Should I remove the drywall and check everything?

I'd try to find the spot (or area) the water's originating from and work from the bottom up. If you're lucky, the water only went behind the wall and into the carpet and is done. Check for signs of longer term water (and mold, etc). One of these might help you figure out if the moisture content is higher in one spot than in others:
https://www.amazon.com/General-Tools-MMD4E-Moisture-Digital/dp/B00275F5O2/ref=sr_1_2?ie=UTF8&qid=1466430705&sr=8-2

Thanks again for the advice. It's been a while (again). I guess I move slow. Things look pretty dry. I peeled off parts where the paint was bubbling:

I don't know if there is anything there I should be worried about. After checking out your link, I starting reading up on moisture meters. I see that there are ones with pins and then surface ones. Would either work to see if there are any problems behind the wall?