Hah! I figured everyone would want 1080p at least these days. I used to do the same. It's far easier than figuring it out by hand.
I suppose you would cut the background on a grid laid out in Photoshop/GIMP. If I recall correctly there are a few old templates floating about that have different resolutions marked out in a grid already that you can use as an overlay, but this is still needlessly tedious work in my opinion.
As I banged my head against the wall trying to remember and explain this stupid process I decided to see if I could figure out a nifty way to do this all automatically for any resolution.
So I did it! You'd just drag the background you want onto the .exe or specify its path in a cmd window. I tested with a few different resolutions (including yours) and it all gets perfectly cut up & converted into whatever amount of .tga's are needed, placed in a resource/background folder. It also of course writes to BackgroundLayout.txt and then you'll have a GoldSrc background ready for use!
I haven't tested on any other computers and still have to tidy a few things up if I were to release this fully. But if my madness somehow produces something that truly works, I imagine this is all you'll need to convert backgrounds in the future. Hurray!?!