I just had some vinyl from the floor tiles in my basement tested and the results were positive for asbestos. Also, the black mastic was found to contain asbestos. So now I totally avoid walking in those areas of my basement. I, too, want a permanent solution ASAP. I won't put carpeting down, but since the tiles are warped/buckling, I cannot just lay new vinyl flooring on top of the warped areas because the new floor will not lay flat in those areas. Yet if I remove the asbestos-tiles, I'll have to minimize risk of exposure to asbestos as best as possible, and I'm not an expert by any means...and I feel weary like you do about a possible hazard.
If you were planning to take up those tiles yourself, make sure you use a face mask/respirator approved for asbestos removal just to be on the safe side. You may want to cover the walls and other objects in the area of removal with disposable plastic sheets for protection. You may even wear clothes that you could dispose of, depending on how much dust you got on them. Also, keep the tiles wet when removing them to minimize asbestos being released into the air. And place tiles in a sealed bag with "Asbestos" written on the bag.
Products like floor tiles contain nonfriable asbestos, which means asbestos fibers are tightly bound. If tiles are left undisturbed and if they are not deteriorating, floor tiles pose no real risk. However, the nonfriable asbestos can become friable, and therefore airborne if floor tiles are sanded, cut, scraped, disturbed.