EUREKA SPRINGS, Ark. — The Rev. Randall Christy was sitting in the amphitheater of “The Great Passion Play“ Friday afternoon, two weeks before opening night. Below him, amid a stage set meant to evoke ancient Jerusalem, a cast member in a baseball cap took Pontius Pilate’s horse-drawn chariot on a practice run.

But Mr. Christy was speaking about more contemporary troubles nearby, in the gay-friendly little tourist town over the hill.

Mr. Christy and his allies here contend that Christian visitors are the foundation of the tourism trade in Eureka Springs, a Victorian-era spa town of 2,100 residents in northwest Arkansas near the Missouri border. For nearly five decades, those visitors have flocked to the Passion Play, the seasonal outdoor depiction of Jesus’ final days that is presented in this 4,000-seat amphitheater. But in recent years, Mr. Christy said, those tourists have become more reluctant to visit Eureka Springs because of efforts to promote the town as the “Gay Capital of the Ozarks.”

“I think that is a mistake,” said Mr. Christy, the pastor of Union Valley Baptist Church in Ada, Okla., and founder of a gospel radio network who has operated the nonprofit Passion Play since late 2012. “Family vacation destination should be the thrust of this town again.”