Test for asbestos. If it's positive, stay all the way out of the basement until the problem can be eradicated. Lung cancer isn't pretty or frugal.
If it isn't, buy a good air cleaner, set it up, spray the basement with chlorine bleach to kill mildew in the dust, and then clean it up when it dries. Not a pleasant job but the electrostatic cleaner should keep the dust fought back, once you get the first and worst layer out.