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?
Building a dog house yourself is one of the smartest investments you’ll ever make, OP.
Not only will you have a far more attractive place for your dog to stay, you can feel proud in the knowledge that you built your best friends home with your own two hands. Click here to check out some insulated dog house plans that are cheap.
I highly recommend these. They helped me put up a dog house in my garden rather quickly — and they're insulated too!
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.