Dish soap with a little bleach (read label to be sure they can be mixed) to kill the mold and get the dirt off; then oil it with "Leather Therapy" which has a mold inhibitor. Then store in a DRY place and re-clean regularly!
If you get a jar of Stubben's glycerine saddle soap, it lasts forever and does a good job of stripping nasty stuff and waxes without drying it out. I would wipe the saddle really well with a damp rag, then soap it, then condition it or oil it. I don't really use oil anymore, but Hydrophane was great. I just ran out of an enormous jug that lasted for years! Now I use a leather conditioner called Leather Keep, sold by an australian saddle company. It's really nice; I love it because it really gives the leather a supple feel without making it too soft like over-oiling can do, and it doesn't darken my tack. If I want to darken something, I use olive oil.
Never heard of the vinegar thing! I'll try it next time, I have a western saddle that attempts to become a chia pet every winter.