This started around 04/25 on both my and my wife's computers. This leads me to believe it may be some sort of general problem. Normally, the 1Password icon in Safari produces a dropdown that can be used to log into web sites. In this case clicking the icon does nothing.
We have been solving this by closing Safari, starting 1Password and selecting Help->Troubleshooting->Restart 1Password mini. Closing 1Password and opening Safari usually solves the problem.
I said mini "dies" but I don't actually know if that is what is happening. It may just stop communicating with the Safari extension. Some of the stuff I have read suggests that if mini actually shut down I would not be able to start 1Password.
At the time we were running 6.6.4 but have upgraded to 6.7.0 and 6.7.1 without the problem going away.
1Password Version: 6.7.1
Extension Version: 4.6.4
OS Version: 10.12.4
Sync Type: Dropbox