Just a reminder - every satellite crosses the equatorial plane twice for every rotation around the Earth. Majority of satellites are below the geostationary orbit. So, given enough time every satellite will hit the tether - and even if tether is near-zero width, the satellite isn't.
One of ideas was to dynamically move the tether - from the Earth, making propagating wave along the tether, which would avoid the satellites. This means we need the good means of watching the space and predicting satellites movements - even small satellites can bring trouble. But that's not enough, we might get a situation when we need to perform conflicting movements, considering the current state of the wave.
This problem is well known for decades. Attempts to launch satellites from the elevator - and the elevator makes it tantalizingly cheap, so we'll want to launch many - will exacerbate the problem. The whole world astronautics will have to adjust to accomodate the elevator - and nobody knows how it is going to look like.