Killing Skunks
If it's legal in your local area to kill a skunk, go ahead. Be aware that local law enforcement may have a problem with you disharging a firearm on your property, so check those laws out too. Trapping is popular for states that allow it, as well as body grip traps that kill on contact.
Shooting a Skunk
Any small-caliber firearm will be plenty to take a skunk out, but shooting wildlife is always more complicated than people realize. There are assuredly local laws that govern the use of firearms–even on private property. In addition to those laws, the skunk may not be currently in season, and you could receive a fine for gunning it down out of season. Oh, and you might have some laws about disposing animal carcasses in your area.
Woops, forgot to mention that poor marksman often leave the animal injured, which can be considered harassment of an animal, in addition to the bloody mess they'll leave as they run off to go die under your house and stink up the place. An injured skunk will also likely spray in defense so there's that to deal with too. Geeze–this is more complicated that you thought, right!?
Don't Shoot a Skunk, Trap It
Body grip traps work well on skunks and might prevent them from spraying if they sever the spinal cord before they get a chance to spray. The only real benefit to using traps is that you ensure you won't get sprayed, but the skunk spray can linger on grass and in the air for a long time due to its oily gassy nature. Household pets may go exploring and roll in it, cover their pawas in it, and track it in your house, which can be terrible. Body grip traps also require expertise and skill to both set and maintain. If you are not trained to use these traps, do anot attempt to do so on your own as they can be very dangerous.