Broadway increasingly relies on celebrity casting, but this year, with Hugh Jackman, Emma Stone, James Franco and other A-listers on stage, theaters have been spending ever more time cracking down on shutterbugs, hecklers and the throngs of fans at the stage door.

The scrums that are now nightly, traffic-blocking headaches for security and police “only used to happen with, say, Burton and Taylor,” said the longtime theater director Gregory Mosher.

“The River,” a solemn play starring Mr. Jackman, better known as Wolverine in the “X-Men” movies, draws so many star-struck, picture-snapping attendees that the production resorted to sending out an understudy before the show to remind them to turn off their phones. In November, Time Out New York published a guide to what not to do at the show, including “Applaud Hugh’s entrance” and “Clap after every scene.”

At one performance, when Mr. Jackman asked his co-star if she’d like whiskey or wine, an answer came from the audience: “Wine!”

A representative for the show declined to comment.