If the break is below the half-way point it will go up if the break is above the half-way point it would come down.
Also the process that seems to most likely for construction would be to deploy from the mid point of the cable and then spool in both directions at once. This way the overall forces remain in balance.
Also everything I have read about planned space elevators has it based in the middle of an ocean. This allows some movement if necessary to avoid something large in space but also gives some safety in the event of a failure. If the cable broke below the mid point controlled explosions all the way along the cable would reduce most of the damage with the a large % of the resultant bits burning up in the atmosphere or landing in the ocean. It would probably be much worse for things in orbit than things on the ground.
Taking space elevators to their logical conclusion though would see them being the bases of super towers that reach into space. The cables end up being the foundation supports of the tower.