If you're not using stone or ceramic, here are three sound choices for the subflooring. 1) Delta FL or equivalent followed by plywood or OSB. 2) Dricore or equivalent; 3) XPS (or EPS) followed by ply or OSB. The finished floor goes on top. Spring to fall, run a dehumidifier and check relative humidity levels if you're not sure of how dry "dry" is. Keep the windows closed.

When your budget permits, remove the drywall and start over with EPS, XPS or foam on the walls. No fibrous insulation or vapour barriers in the basement walls!

Thanks for the input.

I failed to mention that we have ceiling height issues, so I dread the idea of adding a lot of flooring materials even though I know it's the 'right' way to do it. Plus, I suspect that all of those flooring options are way beyond our puny DIY budget, which is basically enough for oops paint and closeout carpet tiles or something similar. I really need to know a cheap way to make the floor comfortable enough for kids to play on and not have the space smell musty.

So for now, we're OK with uninsulated walls?

I've used commercial low-pile carpets with no rubber backing on homes I've built myself, which have basement slabs of six inches of gravel followed by four inches of concrete. I'd be leery of using carpet on anything less than that. Why not just paint the concrete and use non-rubber backed throw rugs that you move often?

OK with uninsulated walls? Not much choice, you say. Be sure though to keep the dehumidifier going.

Worthy: Why no rubber backing on the commercial carpet? Does rubber hold moisture? (please excuse my ignorance :) One of the big box stores had the lowgrade carpet with white padding for 66 cents. I like the idea of a little bit of cushion for the kids, but only if it doesn't hold moisture.

I would paint the concrete, but we have that old black adhesive from a long-ago-removed tile to worry about. The lady at the big box store said it probably contains asbestos. I'm guessing that I can't just paint over that. Or maybe I can?? The surface would be very mottled with the adhesive and then the pits from removing the carpet tack strips along the edges. But maybe that would look cool? Would the paint seal the concrete or would the moisture cause it to flake off?

Pressure differentials, capillary action and (depending on the construction) drying pushes water vapour upward into the basement. When you put an impermeable finish on the floor, it traps the vapour, which becomes water, which nourishes the mould.

Yes, mastic before the late '70s contained asbestos. It can be removed safely by following the guidelines (follow link at Question 2) suggested by the Resilient Flooring Institute. Don't just scrape or sand it. The adhesive can also be safely encapsulated.

Here is a link that might be useful: Asbestos Abatement

Small update: I did a 48-hour test of two sealed plastic squares over different parts of the basement concrete floor. We even had a good rain storm on the first day. No condensation on either side of the plastic!

My theory that the musty smell is mostly trapped in the old pad and carpet seems more reasonable since we've never seen/felt any water in the finished area (and the unfinished area is where the sump is, which doesn't go off but in very torrential rains). The previous owners did have several dogs, which I suspect contributes to the smell. And we had a bathroom sink overflow last winter. I thought we ripped out all of the wet carpet, but it may have seeped further into the pad than what we ripped out.

I'm hoping all of this bodes well for putting down carpet (without pad) after we paint the walls and then rip out the rest of the carpet and pad. I've read that olefin is the best fiber for basements and to make sure the backing is also olefin and not jute. Does that sound reasonable?

Worthy: Thanks for the link. I printed it out and looked it over, but I can't seem to find anything about encapsulating it. I'm wondering if I even need to worry about the adhesive if we do carpet because the residue is rather thin at this point (maybe somebody scraped it in the past?). The lady at Home Depot said she just carpeted over her old adhesive.

It sounds like your basement is a good candidate for carpeting without pads. For the best prices, check out the small carpet stores specialising in odds and ends.

Encapsulating is mentioned in the Asbestos Abatement link.

The Fortress Institute says you can encapsulate with a concrete water sealer or latex based paint. Sprayed is preferable, as it doesn't disturb the mastic.

(I hate to think of all the asbestos fibers my friends and I breathed over the years either doing the demolition or working in the gutted homes later. I remember one woman renovator who had an entire first floor of mastic sanded off hardwood. In some places it was an inch thick. I guess the floor guy got the worst of that.)

I just helped my sister finish part of her basement. We put in Delta FL + OSB. It ran about $1/sq ft for materials and raised the floor about an inch. HTH in your decision.