How do I cover large gaps in drywall?
Setting-type compound (the bagged kind you mix vs. the drying-type you buy premixed in a bucket) will easily fill a 1/2" gap without cracking.  Make sure you load up the gap well before applying your tape, then finish it as you would any other joint.

Sadly I have a lot of experience filling this type of gap from my own basement project a few years ago...