All my banks send a code to my email. That's not a browser issue. You will need to verify the security settings on the bank(s) site.

Here is what I see at Wells Fargo. Just look for this in the security or account access settings for your bank and turn it off if you don't want the extra security.

2-Step Verification at Sign-On lets you activate an additional layer of security when you access your accounts online by entering a unique Advanced Access verification code sent to your mobile device or phone, or from an RSA SecurID® device.

This issue can be caused by corrupted cookies or cookies that are blocked.

• check the permissions for the domain in the currently selected tab in "Tools -> Page Info -> Permissions"

Detail like websites remembering you and log you in automatically is stored in a cookie.

You can use these steps to make a website recognize and remember you.

• create a cookie allow exception with the proper protocol (https:// or http://) to make a website remember you

You can check that you aren't clearing important cookies.

• using "Delete cookies and site data when Firefox is closed" keeps cookies with an allow exception
• using "Clear history when Firefox closes" to clear cookies clears all cookies including cookies with an allow exception you may want to keep
• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exceptions for passwords and other website specific data

• Options/Preferences -> Privacy & Security
  Cookies and Site Data: "Manage Exceptions/Permissions"