In my experience, "remember me" on banking sites usually saves only your username/login name. Useful on your personal computers when your bank uses your 16 digit card number for login name.
IMVHO, if someone has access to your cookies, you are likely dealing with problems bigger than protecting your bank card number. That implies having access to your files, physical access to the machine, or MITMing the connection to your bank. I can think of worse things that can be done with that level of access.