I have a late 2012 mac mini, I use safari, websites say I need to update safari but there are no updates available. I use high sierra 10.13.6, can I update or not?

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Safari will open a redirect and open the Apple Store where the download can commence.

Does this computer appear as Listed and qualifying for an upGrade to a newer version of macOS? 

The below links will help identify which version of macOS  this computer may Qualify to run.

Unless I am mistaken, this computer may Qualify to upgrade to Catalina macOS 10.15

With Catalina macOS 10.15 comes a Newer Version of Safari

If upGrading is not to the User liking - another link regarding Different Browsers that can run on High Sierra in

link below

Browsers by Mac operating system - Apple Community

macOS Catalina 10.15 hardware requirements

macOS Mojave 10.14 hardware requirements

Make Time Machine Backup  is very useful 

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina, Big Sur, Monterey, Ventura supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may require an update or upGrades to 64 Bit Version

Mostly you just need a more modern Browser than Safari on Mojave, or High Sierra like Firefox or Brave.

https://brave.com/

Thanks to dialabrain…

Firefox 111.0.1 is now the current version. runs in High Sierra.

https://www.mozilla.org/en-US/firefox/new/

FWIW, Catalina 10.15.7 is the latest version you can run. Instead of upgrading you can also use a more modern browser such as Brave or Firefox.

Click on the blue links which will take you to their respective pages.

However, what happens when you open this link?

macappstores://apps.apple.com/app/macos-catalina/id1466841314?mt=12

BDAqua wrote:

Thanks to dialabrain…

Firefox 111.0.1 is now the current version. runs in High Sierra.

Who? What does he know!? 😎

When I click on How to get old versions of macOS the following Apple Website will open to as per image below

When I click on the Blue Link of that link with Safari Browser will open the Apple Apps Store when Catalina will be presented to Download

Refer to Second Image

Second Image

Bank changed its system. (My iMac uses High Sierra 10.13.6) I have been unable to download the new app for the bank. It wants me to update my computer. When I try to download a more recent Safari, it requires me to update my iMac.

I don't know what else to try to restore my ability to access my account on my bank.

:)