Hello Mark, if you are concerned with mice, don't be, you can't confine them to one part of the house with doors or blocking apparent access. Mice can materialize through solid walls!!
Actually they can squeeze through a gap you can't put a pencil through.
Mice go after food, so if there is nothing for them to stay and eat, then they leave. Only in the winter will mice stay where there is no food, and then they are staying to make a winter nest. I find mice nests in my tool box drawers amongst the steel tools and sockets. All because my shop is kept above freezing temps.
As for putting in a half door to the basement that will be almost as bad as keeping the full door closed. The heated air from the basement will rise and try to go through the open upper doorway, but the cool air on the floor of the upper floor MUST be able to go down the stairs and return to the stove, completeing a heating loop. With the lower half blocked, No loop, no circulation, no heat movement through the house, and you will be unhappy.
If your basement is fully insulated but not drywalled or finished, then install a few ducts like the above photos show, but put a return duct in the floor of the upstairs as far from the heat ducts as possible. This will create the all-important heating loop needed to move the hot air. If your basement is fully finished, you may have to use the stairway with fans to move the heat.
Hope this helps.. Greg L.