You could maybe see banks having plaintext communication as an optional, but I doubt they'd make it default (allowing users to switch to html).
Isn't this problem already solved with certificates online? Shouldn't this be solvable the same way? E.g. a bank sends an email containing a link to the content with some special attribute. The web browser displays the content if and only if the sender domain of the email (e.g. hsbc.com) is also the domain from which the content will be downloaded.