I have an Airport Express Base Station Model number A1084. I think it is 1st generation. Is it any use with Mac OS 12.4 and brother wireless printer?

The A1084 is obsolete, since it requires an old version of AirPort Utility that unfortunately will not run on any Mac or iPhone / iPad that has been offered in the last 10 years.

However, if you have a PC.......you might be able to download and install AirPort Utility for Windows.....which might allow you to configure the AIrPort Express that way.

Download AirPort Utility 5.6.1 for Windows - Apple SupportOfficial Apple Support › ...

The ironic thing about the A1084 is that you might be able to get it working with a PC, while you won't be able to get it working with an Apple product.

AFAIR there was an utility that could wrap the old AirPort utility.app so it could run on newer unsupported macOS versions so the user could configure old unsupported AirPorts (like my AirPort Extreme 802.11g Snow Dual Ethernet Base Station 2003 M8799K/A, for example).

I also have an old G4 PPC PowerBook running Mac OS X 10.5.8 Leopard that can access that obsolete AirPort.

But I have retired that old AirPort anyway because I do not want obscure errors in the household network.