Fix wall near bottom of staircase
I would cut a thin strip of drywall to fit that missing section. It will be a little awkward because it's thin and long and therefore easily breakable. Cut it from a piece that has a finished edge to help hold it together. That edge should face the door frame. Because it's so narrow the best way to fasten it in place will be with construction adhesive. You may have to support the strip from behind to get the strip level with the existing drywall. Scrap cardboard should do the trick.
When mounting the strip pack some spackle between the strip and the drywall. The reason you want to do this is because the repair will be too narrow to tape and the packing will help keep the spackle from cracking.
Smooth any excess spackling and hold the strip in place for a few minutes to let the adhesive set.
You can then finish the job with additional spackling and sanding as needed after the adhesive has set firmly.