You can give it a go if you'd like too. Drywall is one of those things that is easy to do but is a true art to get it done right. If you want to give it ago it won't cost much but will take some time and rework to get it to the quality of those new stairs. A dry wall crew will get it done on the first pass and it would look great.

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Looking at the picture again, you could cover the area with a nice piece of wood that matches closley to the stairs and floor. This might be alot easer DIY wise.