From my research, both exercises - box jump/vertical jump and the stair climber - generally activate the same type of muscle regions (legs as well as core) on top of strengthening the cardiovascular system. Besides box jumps being an explosive movement are there any other additive benefits of box jumping over the stair climber? What are some alternative exercises to these exercises since they both seem to be pretty harsh on the knee joints?
There are different ideas about jumping exercises. I cannot show any references right now, but some people believe doing explosive work increases explosiveness; whereas some people only believe that increasing strength is the only way to increase explosiveness.
I believe that one should work out as needed. For example, do you need to do box jumps? Do you need jumping in your daily life? Do you need to get better at jumping and jumping higher? If not, I do not believe that doing box jumps would bring you extra benefit than regular strength training.
However, the risk of injury is higher with jumping, which means you need to be more careful with it.
Explosiveness= strength+ speed
Speed is all about type II muscle fibers and strength is about progressive overload. Using weights trains both type I and type II muscle fibers, higher intensity trains both in a collective manner and high volume trains both in a cumulative manner.
Training for endurance or strength will passively improve your explosiveness and speed, many olympic runners use weights to train increase their speed.
If you want to jump higher normal weight training or calisthenics are enough, if you really want an incredibly elite level jump then treat jumping as a skill.
To get better at a skill you practice it more.