I am also having this issue. The Chrome extension worked fine for a day after it was installed but, once I rebooted, it claims "Avast Passwords is not running", even though I have the Avast Password window open and logged in next to my browser.
OS: OS X Sierra (10.12.3)
Browser: Google Chrome (57.0.2987.133 (64-bit))
Avast Passwords: 1.4.2203
Avast Passwords Extension: 1.1.2746
I have tried removing and re-installing the extension but to no avail.
I'd appreciate it if you could tell me how to resolve this issue as I was hoping to purchase and roll-out Avast Password across my company but I need to know it's reliable first!