Long ago, the Catskills were a punchline: kitschy motels on Route 28, faded Borscht Belt palaces, the ghost of Dirty Dancing haunting every highway billboard. That era is over. The past decade has seen a transformation so thorough that luxury hotel groups—Auberge, Hyatt’s Unbound Collection, and soon Soho House (on the east side of the Hudson)—are now staking serious claims in towns like Livingsto...