Safari

You may want to check with your bank customer support or their website regarding their system requirements. If they don't support using Safari, they likely do support Firefox. You can have both installed on your Mac at the same time. https://www.mozilla.org/en-US/firefox/new/

If it is supported and still doesn't work, try setting up another admin user account in System Preferences/Users & Groups to see if the same problem continues. Please post back on whether or not this worked. Also try the Safe Mode. Please post back on whether or not this worked.

Isolating an issue by using another user account 

Safe Mode - About

If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor.  Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown. If the log won’t post, try posting it in Pastebin and provide a link in a reply. After pasting the report in a PasteBin page, go to the top of the page, and copy the address in the URL bar. Paste that in a new reply.        Pastebin

Etrecheck – System Information

That's fine. You can continue to use Safari for everything else if you prefer or whichever one suits you.

That's fine. You can continue to use Safari for everything else if you prefer or whichever one suits you.