Unfinished leather will spot and patina over time. Expect that the leather will spot or darken where lube or body fluid touches it. Finished leather (or garment leather, usually with a slightly glossy surface finish) on the other hand, is less likely to spot or darken when moistened with lube or body fluids.
If you want to clean your (unfinished or finished) leather strap-on harness, use quality saddle soap (specialized leather cleaner you can find at shoe repair shops) and a dampened cloth or sponge to clean. Lightly rub the cloth/sponge over the surface of the saddle soap to create a lather and rub said lather into the leather. Feel free to use a bristle brush to get into the rivets and crevices.
Then, rise and squeeze out the cloth/sponge and use it to wipe away the lather. You don’t have to rinse the saddle soap off. Let it dry fully by giving it a quick pat down with a towel, then hang it in place with good ventilation.
And remember: don’t put it near a heating source! Store your harness in a cool, dry place, out of direct sunlight. Hanging it up is ideal, but pay attention to things that might scratch the surface of the leather.
You can condition the leather after cleaning or when leather feels dry to the touch, and make sure to spot-clean with a damp cloth before drying thoroughly to store. Leather is skin, it requires a certain amount of oil to stay soft and supple.
Keep in mind that leather is an organic and porous material and as such it can become home to mold. Which means moisture will encourage tarnishing in the hardware. It can’t be sanitized with harsh chemical cleaners either, not without doing damage to the integrity of the material.