Use half white vinegar and half water. Wipe down all leather thoroughly, then wipe down using a clean rag and clean vinegar solution. Let the tack air dry.
If you have suede, put the solution in a spay bottle and spray the suede. Use a clean rag to remove as much of the mold as possible. Then use a horse brush or suede brush to "fluff up" the nap.
DO NOT use standard conditioner or oil on the tack. The only thing I have found that works to condition tack and not attract mold is Passier Lederbalsm (and I'm in Arkansas, mold capital of the Nation, lol). The beeswax in the conditioner helps seal the leather. Use your hands to put it on, rub it in well (warming the conditioner), and let it set for 15-30 minutes. Then wipe the tack down thoroughly with a clean rag and buff to a glow. It's important to remove the excess conditioner.
Keep some white vinegar and rags in your tack room, and wipe down any mold you spot IMMEDIATELY. If you keep after it, you can keep it from getting bad.
Putting old towels over your saddles and bridles will also help. They will absorb the moisture in the air before it reaches the leather. Swap out with fresh towels every 4-7 days.
I put leather tack that I don't use in one of those giant zip-lock bags (used for bulky storage). Keeping it air-tight greatly reduces the chance for mold to grow.