This is most likely not a Firefox issue. Most banks that I know actively disable the possibility of saving these details for security reasons (which is stupid, I know). If you would like to be sure, please tell us which bank are we talking about.

Details about a website remembering you are stored in cookies.

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 the website remember you
• use "Delete cookies and site data when Firefox is closed" to let cookies without allow exception expire when Firefox is closed
• Options/Preferences -> Privacy & Security
  Cookies and Site Data: "Manage Permissions"

In case you use "Clear history when Firefox closes" or otherwise clear history.

• do not clear the Cookies
• do not clear the Site Preferences

• Options/Preferences -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings
• https://support.mozilla.org/en-US/kb/delete-browsing-search-download-history-firefox

• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exception for password and other website specific data
• clearing "Cookies" will remove all selected cookies including cookies with an allow exception you may want to keep