I don't think you can make this area functional. Ideas to dress it up: a runner, wallpaper, a hanging fixture, a picture on the stairwell's wall. Functionality in the closet by adding closet organizers, boot racks, hooks, etc.

depending on how necessary it is to keep the door on the closet and how tidy you keep things...

(your dog is so dang cute!)

I like Maureen's ideas. You could also hang artwork on the coat closet door. Something that would cover the upper portion of the door. One great piece that is simple-example: artwork on canvas that wraps around the canvas versus framed. Google "hanging artwork on a door"