Unity uses NSPasteboard (OSX shared manager for clipboard) to hold the data for Drag&Drop, so any process that monitors/modifies the clipboard may be messing with the Drag&Drop.
I think there might be a Chrome extension that would be generating notifications that would for some reason break drag and drop. Disable all your extensions in google chrome and if that doesn't solve your problem in Unity then do a fresh install of google chrome.
Edit: As suggested by other users, another solution that might work is
Quit Unity and force quit the pboard process via Activity Monitor and
then launch Unity again.
Hope it helps.