Take this as the usual unreliable Internet advise because I admit to not being an expert in Internet security.
I think the biggest concern you can have with private and public stuff in the same tabs is accidentally entering private information into the other tabs (e.g. copy pasting some information between tabs by accident). There's always the risk that one of the "bad" tabs you have opened has just put a virus or other malware on your computer which is now tracking all your actions, but that can happen whether you use lots of tabs or not. Unless I'm mistaken, the only way information can "leak" between tabs is you, malware, [EDIT: bugs], or another compromised piece of hardware between you and the bank leaks it, because the tabs are simply displaying what you've downloaded from different web pages at once. The other thing to note if you use lots of tabs is that you may forget to log off and therefore leave your banking or other private session open for longer than it has to be.
I don't trust Chrome enough to use it for anything other than Google services

I don't use it as a browser - more like an interface to gMail / Google Docs / etc. At least when using it for those services I already know my every action is being tracked by Google and Chrome doesn't change anything.