You should not try to remove them yourself. If you want to just go with a quick and basic remedy then covering of asbestos tiles is an acceptable way to abate the problem. Covering should be enough to prevent and more damage or wear and tear. Carpet with a good pad is always a good idea (in a basement like yours, go with a rubber padded carpet pad rather than a fiber one, and maybe consider installing a product like dri-core to help prevent dampness) You could also go with a new vinyl or linoleum right over it depending on the condition.
If you are concerned about the condition of the floor or want to just have it removed you should contact a professional Asbestos abatement company and make sure they remove it safely and legally.