I've had many, many problems with the Evernote Web Clipper for Safari. The most recent problem was that Evernote appeared to be logged in but each time I clicked the Evernote Web Clipper icon, a new tab would open, "Success!" would quickly appear in the title bar, and the tab would close.
The most recent fix for me seemed to be deleting all saved passwords:
Safari > Preferences > Passwords > Select All > Remove
I had previously removed just the Evernote passwords but that did not seem to help. I had also emptied all caches many times without success.