Some of my banking apps are not working on my iMac

Chase Bank and Duke Energy need to stop using the outdated and privacy-intrusive "user agent" query as the means to determine if a browser can handle their website, which was never a good idea to begin with.

If you are motivated to do so direct their attention to this excerpt from Apple's Safari Developer Documentation (emphasis added):

Important

Use feature detection to enable or disable functionality instead of relying on the name or version of the browser in the User Agent string. Learn how to detect available features on MDN…

The passage regarding feature detection in that document uses even stronger language:

"... this is a terrible practice that should be discouraged at all costs."

Unfortunately, continued and inexcusable use of that terrible practice leaves you out in the cold until they realize the error of their ways and correct it.

The workaround for end users who are unable to update their Macs to a newer macOS version is to use a different browser.

Consider Brave, or Firefox:

https://brave.com/

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

The bottom line is that there is nothing wrong with your Mac's version of Safari, but those websites are using an outdated and ill-conceived method of determining browser requirements.

If you do "About this Mac...'" from your Apple menu and share the iMac SUB-MODEL, we can tell you if you have an OS upgrade path available.

The "About" window looks like this:

Please include any modifiers to the year, like Early, Mid, or Late.

⚠️ Please DO NOT POST the serial number for security reasons.

However, I also approve of using FireFox. It still supports macOS 15 and will be a much faster solution.

As noted above, Safari cannot normally be individually updated. Even when it is you may still have difficulties connecting with some sites. (Chase Bank comes up a lot here as a problem.) The solution is to install and use another browser for these problematic sites such as Firefox which causes no problems.

Safari versions a tied to the macOS version and can not be upgraded independently of the macOS.

Your best bet at this time is to get and use a third-party browser like Mozilla FireFox.

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

Another option is simply replacing the Mac, if it's so old that you cannot open your banking software it may be time to replace it. You have a LOT of options available to you for that if it's something you are considering.