macOS Catalina is new and it breaks a lot of old software and as a result, you are not unwise if you choose to forego it for a while. (My Mac mini server is happily staying on Mojave.)
My old pal Rob Griffiths has the solution to stop Apple from bugging you:
Apple really wants people to update to Catalina, so they let you know about Catalina…constantly, it seems. In System Preferences > Software Update, you’ll see [an enormous upgrade box]…. And while that’s annoying, it’s not nearly as annoying as the red “1” dot they stick on System Preferences, which will stare at you forever. I complained about this on Twitter, and as is often the case, some very bright people had solutions to the problem.
This is more involved than it really should be, but at least it works.
—Linked by Jason Snell