The banks can choose whether or not to support a browser, and then specify the level of security.
Up until I swapped to Firefox, Mozilla always was rejected by NatWest with a msg saying 'You are using an unsupported browser'. However, it did work with Cahoot (Abbey National's Online bank), which won't work with less than 128 bit encryption.
Basically, check with you bank, or download and try it. No bank system will let you in unless the encryption is working as it should and at a high enough level.