Asbestos is considered to be benign as long as the material is not friable. The United States EPA states that asbestos flooring in good condition is generally considered to be safe but the materials can become friable with age, or by sanding, grinding, or with demolition. You would need to consult an licensed contractor that has experience working with asbestos tiles but in general, the Minnesota Department of Health has a great resource on residential removal of asbestos tiles: http://www.health.state.mn.us/divs/eh/asbestos/floortile/index.html
There is no amount of asbestos exposure that is considered to be safe. Asbestos related diseases are dose-dependent with increased exposure resulting in a higher likelihood of disease – but not everyone gets sick. In addition, the diseases have a latency period of 20-40 years from exposure to potential onset of disease. Shipyard workers and construction workers with asbestos related diseases generally have been exposed high quantity of airborne asbestos for many years.
In your case, removal of a basement full of asbestos containing tiles does not turn your home into a health hazard – especially if the tiles were in good shape and you didn’t sand, grind, or demolish the stones. While each person’s susceptibility to disease is unique, there is a low chance of developing any asbestos related diseases from the minor construction you describe.
You should consult your family physician but from what you are describing, no medical testing is warranted at this time. However, it may be beneficial, for peace of mind, to avoid using your basement until you test it for airborne asbestos and wash any potentially contaminated clothing/materials.