Yes, and that force will result in the ribbon coming to the ground at a greater angle (wind resistance will mean it won't be perfectly vertical even when not being climbed), producing additional horizontal force components at its mooring, which will opposite in direction to the earth's rotation, bleeding off a little of its angular momentum. There's no free lunch, and since the elevator system itself, once in place and stable, has no source of additional energy (like thrusters on the station or counterweight), it must come from the earth itself.
Horizontal AND vertical forces at its mooring AND at the point 62,000 miles out, lowering the orbit of the endpoint and shifting its location. You could end up "cracking the whip" with that endpoint. If there's a mass out there to provide tension on the ribbon, you could lose it.