MOUNT PUNTANG, Indonesia—Surrounded by volcanoes that make the soil here some of Indonesia’s most fertile, members of the Klasik Beans cooperative sniff, sip and swish tiny cups of coffee like a sommelier would a coveted vintage.

The motley crew, many of them search-and-rescue volunteers, talks proudly about its efforts to produce the perfect flavors and cultivate new varieties. Klasik, which specializes in rare, high-end coffees, is known for its Sunda Hejo, a rich, forest-flavored Arabica that has put it on the map among...