Of all the practical problems facing a space elevator, dealing with debris is by far the easiest of them.

You don't run one cable from the ground up to orbit; you run several of them parallel to each other, spaced far enough apart from each other that no piece of debris could sever more than a certain amount of them at once. And at regular intervals down the cables, they'd be linked to each other such that if any subset of cables gets severed, the remaining cables would continue to hold the entire structure upright.

How many cables you'd need and how far apart they'd need to be would need to decided upon by dedicated research into the nature of the debris problem -- how much debris, how big it can be, etc. And then you just engineer in redundancy for the unavoidable failures to reduce their impact into being a bothersome maintenance task to repair/replace severed cables rather than a complete catastrophic disaster collapse.