First, do some in depth, online research on the subject. This should put your mind at ease when it comes to how stable and safe asbestos floor tiles are when left undisturbed. If you're to the point that you are covering a broken tile with plastic, you really need to get a better understanding of the hazards (or better yet, lack of hazards) involved.
Second, there is no way I would waste the time, and money to remove it. I would go over it with another layer of vinyl, either 12" tiles or sheet vinyl. Modern versions of the tile you have are known as VCT, or vinyl composite tile. They are stocked at the big box stores, and are extremely durable, and not too difficult to install. It' just a matter of spreading the proper adhesive on the existing floor and dropping the new tiles in a straight line. A lot of the typical choices look like something you would find in a grade school hallway, but who cares, it's in a basement. The other option would be more contemporary designs available as "self-stick" tile. The only drawback with these is that they don't really stick too well sometimes, so it's best to spread a thin layer of the proper adhesive on the floor, before you peel and press the new tiles. This would not only solve the problem permanently, but as a DIY project it should be at least $2000 cheaper than removal.
o