All climbers should do push ups. Period. Well, that is my opinion... I have some arguments to back it up though that may sway the many climbers that dismiss push ups as a useless exercise for climbing:
- Pushups work the triceps and pecs: important antagonists for the pulling muscles that we use in climbing moves. Keeping the antagonists just as strong as the pulling muscles helps keeping a balanced, injury free body. But it doesn't stop here: both the triceps and the pecs are recruited in many actual climbing moves: when moving up one hand, we (should) push down with the other. The triceps does this. And think of all the moves that require the pecs to come into action: sidestepping, locking off sidepulls, compression moves, etc.
- Pushups train core strength and stability, especially when instability is introduced by doing the excersice on one leg, using rings, a suspension trainer or a fitness ball. Core strength is required in nearly every climbing move, especially when the terrain gets steeper.
- Pushups can train a variety of shoulder muscles by varying in hand positioning (wide, narrow, above the shoulders, close to the waist, etc.) and by introducing instability to the hand using gymnastic rings, a suspension trainer or a fitness ball. Shoulder injuries are common among climbers and can hold you back for months or longer. Chronic shoulder injuries are almost impossible to get rid of. Strong shoulders are much less likely to get injured. By varying the push up excercises a lot of shoulder muscles can be trained, including all the small, tiny ones that get tweaked easily (think rotator cuff...).
There are endless variations of push up excercises. Varying them is important. Which ones you like comes down to personal taste, but I would recommend introducing some sort of instability and doing only exercises of which you can do at least a few good repetitions. It is easy to injure the small stabilizing muscles with an exercise that is too tough. That kind of defies the purpose of the exercise, wouldn't you say?
For inspiration I've made a little video of the excercises that stuck with me and keep coming back regularly in my trainings. Enjoy training!