This is not the ability of climbing muscles to work aerobically, but the general fitness of your heart and lungs. As this is the engine that helps us drive the rest of the body through maintaining a well-oxygenated supply of blood throughout the body. Whilst there are few studies that have linked high levels of cardiovascular fitness directly to climbing fitness/ability, there is certainly anecdotal evidence that would suggest having a strong heart and lungs can only help. In sports science, this measure is referred to as your VO2 Max. It can be measure directly through expired gas analysis in a lab or indirectly through bleep tests.
Cardiovascular fitness throughout the whole body requires exercises that work the major skeletal muscle groups like the upper body and legs. As such things like running, cycling and swimming are common ways to train this. With the key being trying to keep your heart rate at an elevated rate for 30 minutes or longer. To maintain a set level many people trying to improve this will a run or cycle to a pulse rate or speed.