Throw a tarp over the trap carefully. In fact, I'd throw the tarp over the cage before baiting it. Have it all set up for the possibility. Skunks won't spray in a enclosed container. They don't like the smell any more than anyone else. I don't know your usual method of dispatching predators so what you do from there is up to you. A bullet through the lungs is one method recommended to prevent spraying. Spine shots and head shots usually result in a big stink. Drowning them in a garbage can while still in the trap is another way to dispatch with minimal stink. The skunk may still spray, but the water will contain most of it. Or you can just release them in situ while the trap is covered by the tarp and try again for your target species. You are not specifically targeting skunks I gather from your post, so just releasing any captured would probably be your best option.
Good luck.