>Almost EVERY TIME I log into my on-line bank account I see the following message (popping up from the bottom left tray) (see attached).
>Clearly the AV Avast wants to sell me more of something.
>Can they see my user ID and password? Or they only "know" that I am connecting to a secure site?

To answer your question, yes, Avast can watch everything you do, it has system level privileges (even more privileges than admin). You have to decide if you trust it; if not you need to remove the product.

That particular message may be an effort to get you to use Avast's so-called "SafeZone" browser rather than your regular one. If that's what's happening it looks like it's possible to uninstall just that browser: http://techdows.com/2016/02/uninstall-or-remove-avast-safezone-browser.html