That might work if there was only a single oscillation traveling along the length of the elevator. But, that won't be possible, as each launch (and many other things) would create harmonics if they're not a totally exact counter.
Adding elastic elements wouldn't help (if the would even be possible), dampers would have to be extremely massive and so aren't feasible. Which leaves adding rocket engines everywhere to dampen the vibrations. And control centers, fuel pipes, etc.
So, a cable in the form of a simple strand wouldn't work in any case, unless it's extremely stiff and not anchored.
Btw, you could move elevators up and down with impunity, as long as they have the exact same mass all the time. But people walking around would create vibrations, which would build or reduce other harmonics.