Establishing a sea route over the top of the world has been a centuries-old navigation dream, with explorations in search of the famed Northwest Passage often ending with stranded sailors being forced to eat their boots—and in some cases, each other.

Now, thanks in part to climate change, polar expeditions are becoming the latest thing in high-end tourism.

On Aug. 16, 2016, the Crystal Serenity set sail from Alaska on a historic voyage. With more than 1,000 passengers, and...