I have Osage orange recurve bow that a friend gave me years ago. I was worried about the wood drying out and cracking. What is the best way to care for this bow?
Just unstring it when you're not using it, don't stand it on its tip, and keep it out of nasty weather.
Should I be wiping it down with any kind of oil or something?
Depends on the finish. Most modern finishes do not require protection unless they are damaged.
Wood bows are made to be shot but if this one hasn't been shot in a while exercise it gradually and well before you pull it to full draw.