QUEENSTOWN, New Zealand—Towering mountain ranges, forests and glacier-fed rivers made New Zealand the perfect stand-in for Middle Earth in “The Lord of the Rings” movie series and a cinematic billboard for the country’s natural beauty.

Today, jet boats rip down rivers seeking the mythical Isengard, where the wizard Gandalf was imprisoned. “Freedom campers” in rented vans leave trails of waste. Tens of thousands of helicopter trips annually deposit visitors, some in flip-flops, on New Zealand glaciers that were once the realm...