Part of the decline in the popularity of boxing is to do with reduced exposure as there is a lot less shown nowadays on TV. Can you name one or two of the current World Heavyweight Champions? I expect not. This lack of exposure had probably contributed to the decline as TV coverage brings money to a sport and without this money the sport governing bodies have less funds to use to develop the sport.
Another cause of the reduced popularity is the health risks of boxing. Boxing can involve some heavy blows to the head and this can cause a significant risk of head injury. This could lead to permanent brain damage. The BMA (British Medical Association) opposes boxing primarily because of the risks of injury to the brain and eyes.
Trampolines though are seen as providing a far safer sport and one that is growing in popularity internationally, especially as it is now included in the Olympics.