Do you wash a saddle after each use?

You don't have to, but it is advisable to at least wipe it down with a damp cloth to remove sweat and dirt which are harmful to the leather.You should clean the saddle about once a week if not in heavy use. To clean it you'd need something like Lexol Cleaner, Murphy's oil soap, or a tin of saddle soap, water and some sponges or rags.

No, but when you do, don't use baby wipes it makes the leather crack.

Murphy's oil soap is a good soap to wash your tack with.

To clean a saddle and other tack, you need a bar of saddle soap or liquid saddle soap, a bucket of hot water, a sponge, an old soft toothbrush, a 1 inch wide paintbrush, and neatsfoot oil. Sit on a plastic sheet on the floor, or outside on the grass. Take apart the saddle parts that can be removed, remembering how to reassemble. Wet the sponge and squeeze out all the water, add soap, and wipe the leather until you have removed all the dirt. Use the toothbrush to remove caked on dirt. You may have to get clean hot water a few times. Let the leather dry. Dip the paintbrush into the neatsfoot oil, and slather onto the leather. The leather should turn dark.

Clean your leather equipment at least twice per season, and again before you store it if you are not using it for a while.