My bank requires me to verify my identity, if I choose to use online banking from a different computer.
If I get three login failures, the bank locks my account, and I have to call the bank for a password reset.
Does your login work?
Some bank logins use a popup window for the login, and you have to make sure that popup windows are not blocked.
You also may discover that Safari (your OS X default web browser) is not well supported by the bank, and you should try Firefox for your online banking.