I've got a stairway down to a basement that's wide open at the top and at the bottom. The basement gets pretty humid, so we run a dehumidifier down there, but the wide open entry undoubtedly makes the work of the dehumidifier a whole lot harder and less energy efficient. Putting a door on the stairwell isn't practical for code reasons without major reconstruction. Would a curtain across the bottom of the stairs help? It wouldn't be airtight, but would one expect a solid curtain to reduce air / humidity transfer to a degree that would matter?
If there are HVAC vents and a return in the basement, the air is being mixed by your air handler, and a curtain won't make a difference. If that isn't the case, there aren't natural air currents circulating the air between floors, so again, a curtain won't make a difference.
To the extent there is air mixing between floors, that's probably a good thing for the basement. Basement humidity that migrates into the house air will get removed by the house air conditioning, and fresh air in the basement will reduce the likelihood of a mold bloom.
All that said, the basement is humid for a reason, and that reason should be investigated. It's normal to get a little water migration through the basement walls and/or floor. Basements are also generally cooler, so the relative humidity will be higher. However, it can also be a sign of an external water problem.
Examine the basement walls and floor for signs of visible wetness (there shouldn't be any; if there are, you've got a problem to deal with). But even without visible wetness, any water collecting around the foundation will increase what migrates to the inside.
Check gutters and downspouts to ensure that the gutters are tight, the system is clear of debris, and the downspouts discharge far from the house (at least 8' if possible). You may want to consider downspout extensions, or piping the water farther away. Make sure there is good grading around the house. The backfill around the foundation settles for decades and gets eroded; you need to periodically improve the grading.