The essential fault of this concept is that it violates several basic physical laws. One is the law which says that you can neither generate nor destroy energy, you can obly convert one sort of energy into the other.
Well, if you are on the surface of the earth, just on the equator, you will move at a speed of 462 m/s, which gives you a kinetic energy of 107 kJ/kg. The space elevator concept now assumes that you can increase this energy simply by climbing up the rope while it rotates synchroneously with the earth. On a geosynchroneous orbit, you will move at a speed of 3075 m/s and thus your kinetic energy will be 4,7 MJ/kg. The higher you climb, the faster you get and the greater your kinetic energy becomes. But where does this increase in energy come from?