This is an unusual scenario. Since crawlspaces are completely separate from living spaces, they don't usually develop dust like you described.
There is a possibility that what you're seeing is mold. There are many types of mold, and some can grow on wood surfaces, like framing, and look like white dust.
So the first step is identify whether this is, indeed, mold - and if so, what type it is. I recommend having your space inspected by an experienced professional home inspector. The advantage of a home inspector is that they're not there to sell you any particular service - unlike, say, a waterproofer or HVAC contractor or any contractor who's hoping to land the job. A home inspector doesn't have such motivation, which works in your favor.
Find a home inspector who specializes in mold. Know beforehand, though, that even the most knowledgeable home inspector will need to take a sample of the dust and send it to a lab for analysis. The steps to resolve the problem can be determined once those results come back.