The Kessler syndrome would make it difficult or impossible to put satellites in Low Earth Orbit (LEO) for any real duration.... but it wouldn't pose a serious hurdle to travelling through LEO to send missions further out.

And if a Kessler syndrome develops, it is not a permanent state of affairs... stuff in LEO will de-orbit in decades or centuries- and there are things you can do from the ground to speed up that process. There are groups working on using high-power lasers to speed the de-orbit of debris... There is a lot of overlap between my work and that sort of laser application so I have been talking to folks about that possibility.

A Kessler syndrome WOULD be a catastrophe, but not a permanent state of affairs that would doom a civilization to be planet-bound for eternity.