Communicate

If you see a security problem, report it.
If you notice that some part of the online banking system that you use is vulnerable, or the software is allowing access to information it shouldn't, report it to your financial institution immediately. Banks and credit unions have the highest level of security, but mistakes can still exist. Customer reporting of the problem can insure that a remedy is achieved before a error turns into a problem. Additionally, if your financial institution is sending sensitive information via email to you, ask them not to do so, and have them either post it in a secure portion of your account or provide it over the phone.

Ask about security.
Financial institutions understand your concern, and a good one should be willing to provide any information you need. Your contacting them shows them that you care about security. If you have a question, ask.