Of course there’ll be vibration in a Space Elevator cable. You can’t have a long taught cable like that without it trying to behave like a guitar string.
That’s why you need magnetic dampeners on either ends. As soon as an oscillation starts to build, the de-oscillation computer program instructs the rings holding the cable in place to magnetically shift in the direction of the oscillation, thus causing the cable’s movement to dampen.
As with anything new and wonderful, there’ll be many obstacles – but that’s what us hairless monkeys are so very, very good at overcoming.