For half a century starting in the 1950s, Greek shipping mogul Basil Goulandris and his wife, Elise, lived in a manner worthy of a sun-soaked, Patricia Highsmith novel. They palled around with European aristocracy and flitted among their seven homes in places like Paris, New York’s Southampton and Switzerland’s Gstaad—when they weren’t sailing around the world on their yacht, “Paloma.”

The couple never had children. Instead, relatives say they devoted their energies to amassing one of the world’s best private art collections,...