By accident, I've discovered that uninstalling & reinstalling the Google Chrome browser (which was itself quite fragile), as well as the Evernote Windows client, has improved the stability of both of them. I assume there was some sort of conflict between the two - different versions of .dlls perhaps. Worth noting that I haven't installed the Evernote clipper extension in Chrome yet.
I've raised a support ticket for the Windows client. If I find that it is now stable I'll tell Support about my findings.
For what it's worth, I found that the old 'legacy' windows client was quite stable. It's only the new client, starting from a couple of years ago, that has been at all fragile.