First the basics....for conformation (the "beauty show"),he has to be AKC registered and not neutered.
If his breeder shows her dogs, have her evaluate him and see if he could be a show dog. Show breeders will want to see a good dog of their breeding in the ring, but will not want a dog with their name on it being shown that will not win. A good breeder is your absolute best source of information.
If the breeder does not show, it is not likely that your dog will be "show quality". If she is too far away to evaluate your boy then start going to shows and looking at pictures of show dogs. Read the written standard for basset hounds and compare it to him and to the other dogs you see.
Of couse, there are many other venues you can show your basset as well. Obedience is fun, although you need a lot of patience and a sense of humor to do it with a basset. Agility is an exciting sport for dog and owner. The new Rally-O is a good event that lets you interact in the ring with your hound more than obedience. Tracking takes advantage of his best skill. And field trials and hunt tests give him a chance to do what bassets do best. He can also learn to be a therapy dog and be certified to do visitations, a very rewarding occupation. A couple of other titles he can go for to "get your feet wet" are Canine Good Citizen and Temperament Test.