The important thing to remember about online banking fraud is that you are not liable. Even if you get phished and accidentally give away your PIN or other information, your liability is still capped at $50, by law (assuming you report the fraud in a timely fashion).
The reason banks go to such great lengths to protect your account is not to protect you from loss; it is to protect the bank from loss. Most financial institutions have what is called a "risk department" that examines their online offerings and ensures that their products do not expose them to inordinate risk. They are also required to conform to a ton of regulations such as PCI-DSS and FDIC.
Now, whether your user name information (which in your case contains part of your password) is logged depends on how they have programmed their web site. In the vast majority of cases, logging and auditing records are carefully redacted and secured, so that only limit personnel will ever be able to see whatever it is you have typed in. Things like PINs and credit card numbers are almost certainly redacted before they are logged. And most (but not all) banks treat the user name as private information, meaning that it will get redacted too.
Even if someone gets ahold of your password, your account is probably protected by a second factor of authentication including out of band. And even if someone gets past all of that, high risk transactions, such as external transfers (which is how someone could steal money) are often protected by complex risk analysis and "step up" authentication. Even if a hacker used the right user name, password, secret answer, and somehow spoofed your out of band authentication, the transaction will still be refused or held over in an audit queue if, for example, it has a suspicious IP address, or comes from a browser with an unfamiliar fingerprint, or if the transaction processor detects a lot of transactions all going to one place.
Now... back to your question... should you change your password? Well, it certainly couldn't hurt. However, if I accidentally leaked part of my password into my user name field, I wouldn't think twice about it honestly.