A comfy dog bed under the stairs would be enough for the dog. It would feel like a "den" as dogs are denning animals. This would work whether there is a wall and door or just open. This is all assuming the stairs are on the inside of the home. You would do well to get a sturdy dog house if you are talking about stairs out of doors in the winter. Add a heat reflecting bed mat to help them stay warm outside.
Debbie, here's a link to a bunch of projects that may work for you.
This might help: https://www.hometalk.com/8502739/doghouse-under-the-stairs
Do you have a pic of your staircase?
I bet Hometalk has lots of ideas — try the search bar at the top left for help.
I think I’ve seen a couple ideas on Hometalk — try the search bar at the top left for ideas.