kristinj--sorry to report that the fix did not hold. Two hours later (after closing and opening Safari numerous times) it is requiring login again. The fact that is is same OS and Safari version is very vexing. Should be some logical reason why!!
II just purchased a new MacBook Pro and I am having this issue on the new computer but never had it on the previous MacBook Pro. I went into System Preferences/Security & Privacy/Privace and added Evernote to the list of apps that could control the computer. I then went into Safari and Chrome and signed into the Evernote Webclipper. I closed both browsers and did a restart on the computer. I am signed into clipper on both browsers. Will see if it holds!