I don't want to upgrade my OS from 10.11.6, but I'm getting a message to upgrade software to recognize my iPhone 11 ProMax. What do I need to do?

Accept the update as presented. Your current macOS version is 6 versions down from current. You need to consider updating at some point more so if you want to manage your iPhone running a new version of iOS.

If its just iTunes issuing the message it will just update iTunes and not your macOS version.

You don't need to go to the current Monterey if you don't want to, but if your Mac supports it, I'd at least update to Mojave. It will run 32 bit Apps if you need them.

Link-> How to get old versions of macOS - Apple Support

You would need to check that with the relevant providers of those programs. It is hard to conceive of programs which are still being supported by their developers not being able to cope with MacOS updates (presume we’re talking about MacOS?)

Stay10-11-6 wrote:

Will the upgrade change the version of software? I have specific software programs that will not run on higher than 10.11.6; which will save me thousands of dollars.

If that's the case, then you have a choice to make. Replace/update/upgrade the antiquated software you have that won't run on anything higher than a person of OS X that's 7 years out of date and no longer supported or don't sync your iPhone with your Mac.

You need to upgrade your software.

Stay10-11-6 wrote:

I don't understand why Itunes, which is updated to 12.8.2.3 cannot be updated without updating the OS.

Because later versions of iTunes require a later version of macOS.