I don´t know a single large bank that does not offer its customers a way to use a website to do their transactions. There is unfortunatly no single standard set for bank websites and many companies tend to misjudge the importance of a secure website. The Secure Web Bank website did take a look at websites from US, Canadian and European institutes and check wether those provided a SSL Login page and Two Factor Auth for their customers.
If you take a look at this list you see that all European and Canadian banks but one are offering (most require) SSL login pages and that about 50% of the US banks offer ssl only optional. This is in fact a great risk and could be reduced by simply deactivating normal http access to the login pages (or the complete website).
I would encourage everyone who has his account at a bank that is classified as optional to write them an email asking them to remove that insecure access to their website.

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Is your bank really safe?
If you use online banking, security is a major concern. SecureWebBank.com took it upon themselves to database online bank security.
In their own words:
Obviously, this web site is not affiliated with any particular bank or financial institution. The …