Check with the Lowes/Home Depot places, but I think it is called Green Board, and goes inside behind shower enclosures, around bathtubs. Mold and water resistant in such wet settings.
If that is the wrong name, they should be able to guide you to the correct material.
We run a small dehumidifier in the tack room over summer. Just keep emptying it! Makes a great difference in mold reduction on the leather stuff.
We have no sink or water supply inside the tack room. Husband sees a lot of barns with sinks in tack rooms, they all have mold and dampness problems. Even happens if sink is not used daily. So no sink for us.
Look at the 4x8 sheets of siding in Lowes or Home Depot that looks like T111 except it's smooth enough to take paint without any additional work even though it has an embossed wood grain. It's what we used in our dog room and I'd use it again in places like tack rooms. It's much easier to put up and finish than sheetrock and a lot more durable.