I also have an unfinished basement that I'm working on sealing. Generally the dust comes from the unsealed concrete. Assuming your unfinished basement actually has concrete walls and floor, you can do the same to prevent the dust in the first place. They sell concrete sealer at hardware stores and you can have them color it to your liking as well. Just need to wash the floor and walls first, then paint it with a couple coats depending on the brand of paint. Aside from that, if you don't want to seal the basement, you'll just want to create a walled off area using sheets or plastic or similar that will prevent dust from getting into the area and then using an air purifier like you mentioned. This is really only good as a temporary solution, but it could work fine for a quick fix while you figure out your basement situation.

I would wash the floors and walls to get rid of any dust that is present. Then I would make a room out of plastic sheets. Finally, I would put use a 20" box fan with a 20" furnace filter taped to it to clear the air of dust.

If your floorboards are visible in the ceiling you'll want to put up some drywall or something on the ceiling, every time someone walks on the floor above, some dust will get knocked loose and fall from the ceiling. You could put up some drop cloths as well for a quicker, less permanent solution.