Ginko, I was still a bit paranoid about trying to do the job myself, so I figured it wouldn't hurt to have a professional (the same one who removed the duct wrapping) come in and take a look. I also called an company that specialized in environmental hazards in the home as well as speaking to a local university professor who teaches preventive medicine and all of the local EPA's asbestos removal classes. Not one of them expressed serious concern about the tiles. Indeed, the professor was more concerned about potential lead paint in the house as opposed to the tile. The environmental specialist went into a long and involved explanation about how the asbestos is bonded with the tile in the manufacturing process and how difficult it would be to expose it. The contractor, who could have made a mint off me, suggest the same things Glenn and Piffin suggested, or, at the very least, simply cover it with linoleum or new tile (as several of my neighbors have done). He told me not to worry about the tile under the carpet as the carpet would act as a decent encapsulant.
Will I eventually remove at least the exposed tile? Yes, but it has gone way down on my priority list. Is the basement a playroom yet? No, but it is my office. I am not sure if it will ever become a playroom for my daughter, but there is a nice bar down there, so it's already my playroom. I feel as if I have done my due diligence on the matter. Due your own research and see how you feel. Asbestos is not something to be messed with, but on the other hand, don't buy into all of the horror stories like I did. At the very least have a contractor out to look the tile over. It shouldn't cost you anything. Good luck!