First, you MUST stop the source of the mold. Mold required water/moisture, and something to "feed" on. You don't elaborate on if this is a finished or unfinished basement.
I am hoping un-finished. In which case, the mold has little to feed on. Therefore you need to deal with the presence of moisture. If the tile and sump are working, you need to address moisture in the air. You should have a dehumidifer that removed moisture from the air. Keep LOTS of air movement. Run the furnace in the "fan" setting. Also, install a UV light in the ductwork to kill mold spores.
You will need to have a professional remove/kill the mold that is present. I recommend having the house super-ozoneated. This will require you leave for a day or two. You might also consider running an ozone generator in the basement to maintian a mold unfriendly atmosphere.
Finally, consider a water repellant on the basement walls to reduce mositure thru the walls.
You have a difficult issue to deal with, however it is important for your health that you do so.