WASHINGTON—Over 20 nations have curbed the diplomatic or business operations of the North Korean government following a more-than-yearlong effort by the State Department, an indication of the kind of behind-the-scenes pressure the U.S. is using to tackle an emerging nuclear standoff.

U.S. officials have asked countries to shut down businesses owned by the North Korean government, remove North Korean vessels from ship registries, end flights by the country’s national air carrier and expel its ambassadors. At the Association...