Filling a gap between drywall and stair stringer
Wow!  If that's your most important home improvement project, you're really lucky. 

Even if the drywall came all the way down to the stringer and you sanded everything smooth and caulked the corner, I don't think you'd be happy with it. Caulk is very hard to apply right. It looks easy but to get it perfectly smooth and even, Well if you ever caulked a tub or backsplash you know it's not. You could try sanding everything smooth, filling in the crack with drywall mud and taping the drywall down to the stringer. That will probably give you the sharp angle you're looking for. Remember, sharp angles dent really easy.

Personally, I think a new piece of quarter round would look better. Good luck and stay safe out there.