Curtains to direct airflow
A curtain will work nearly as well as a door to block air flow. It will not seal (nor will an interior door unless you weather strip it), but it will significantly reduce the flow going downstairs.
A heavy curtain material will do more to resist the flow of air than a light material will. A light curtain will have a higher tendency to just blow, based on how strong the airflow is.
Simply look for a curtain that is made from a heavy material then hang it low enough that the hem lies on the floor. Alternatively, you can buy a curtain designed for a window that's taller than your door and hang it at the top of the door opening while still having the hem sweep the floor.
You'll probably have to be diligent to ensure that the curtain closes neatly behind you each time you go through the opening because the drag on the floor will probably prevent it from just swinging back to a fully closed position.
If you find that even the heavy curtain you purchased is blowing and not sealing well, use a seam ripper (or small, sharp, pointed pair of scissors) to open up the bottom hem (at one vertical end of the curtain, not the entire horizontal seam) and slide a metal or wooden dowel into the bottom hem to weigh it down.