JAKARTA, Indonesia—When protests against the low-budget, anti-Islam "Innocence of Muslims" video flared across the Islamic world last month, Indonesia's Habib Munzir Almusawa preached a different message to his tens of thousands of followers in Jakarta: Just ignore it.

"If we react so emotionally, then how can we show the good side of Islam?" Mr. Almusawa told worshipers at the al-Munawwar mosque here.