Millions of people have watched a Jan. 11 video by YouTube’s biggest star that included two men laughing as they held a banner that read, “Death to all Jews.”

The man behind the video is Felix Kjellberg, a 27-year-old Swede known as “PewDiePie,” who has amassed 53 million subscribers. His success has brought him multimillion-dollar deals from YouTube and Walt Disney Co., which owns a firm that runs Mr. Kjellberg’s business.

Since...