The issues are huge and probably insurmountable. For example: where would you site it? It has to be equatorial so it can be below the geostationary counterweight. Just the motion of the cable from the eccentricity in the earth's rotation and orbit would probably tear the cable apart (imagine what the foundation of the base station would have to be, and how it would buckle and lift as the earth rotates).

The space elevator is an idea that sounds cool but is fucking dumb if you think about it for more than ten minutes, like the idea of accelerating ships to near light speed, when every hydrogen atom in the void will shred your hull like a cosmic ray.