TOKYO—Anime, Japan's stylized animation that has become hugely popular around the world, helped reshape the country's image as a cultural trend-setter. But behind the scenes, things aren't so rosy.

Japan's animation industry is struggling. Anime workers are unhappy, toiling long hours at low pay. Sales have been declining. On top of that, there is fast-growing competition from across Asia. Studios in China and South Korea now churn out high-quality anime-style programs, helped by cheaper labor and, in some cases, government...