The longtime indifference of staidest Dutchmen to one of Europe’s worst red-light districts has recently been shaken by a series of brutal murders.

In front of the 14th century Old Church in Amsterdam lies a half-mile-square district of gabled houses, narrow streets and tree-shaded canals known as De Walletjes (little walls). An evening stroller, glancing into ground-floor rooms, sees what appears to be a succession of genre pictures by Vermeer: in each, a…