What are the benefits of purple drywall?
I'm not sure. I thought it'd be an easy answer but googling seems to return two different types:


High Density (stronger) wall board
and an updated version of green board (mildew resistant)


It may be that the color is being used by different manufacturers for different reasons. 

In either case, for a basement, I'd suggest using something else. I'd go with the paper-less sheetrock options such as DensArmor

The 'pro' is that there is no paper, so no place for mold/mildew to grow. Instead it uses a fiberglass mesh. The 'con' is that it's fiberglass. So itchy. Be sure to wear long sleeve shirt and gloves while hanging it.