An system is made that lets you vent to the inside of your house. You
have to clean the filter often, and hot air is blown through it. Try
Froogle or Improvements Catalog or similar places.
On another note, you say your basement is entirely below ground and
there is no path to get outside other than through the header or
window. If you don't have a door to the outside from your basement,
you have a major fire hazard.