Do It Yourself

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.