KUNSAN AIR BASE, South Korea — It’s 9 p.m., still early for a Saturday night, as airmen trickle into the mom-and-pop restaurants in this dimly lit neighborhood for bulgogi, ramen and greasy fried mandu. Clusters of airmen stand outside and smoke, and around the corner, where the diners turn into juicy bars, music is already starting to blare. Welcome to America Town, a tangle of narrow streets and neon lights where generations of airmen from tiny Kunsan Air Base have gone to eat, drink, buy souvenirs and meet women. "It’s like a legend. Everybody knows about it in Korea. People...