I'm not sure it does require a different model of access control. It just requires people to actually use the access control mechanisms that exist already.
You should not access banking details or any other sensitive information in the same user-level context as you use to generally browse the internet. The privileges needed for each task ("browse the internet" vs. "check bank statements") should be different. I personally have a separate user account on my machine set up specifically for "sensitive" tasks.
Separation of data access via privileges is nothing revolutionary, nor is it something that can't be done on any modern OS. Unfortunately, online services are still behind. For example, I would probably switch to an online banking provider that let me create one account for viewing balances and another for transferring cash. But these services will get there in time.
User education is a different story.