The proponents of core/stability training state that this type of training is needed in order to improve balance. To a certain extent this is true because balance type activities are included in the core/stability training. Most often, you must balance yourself while you do the exercise.
The balance developed however, is static in nature and thus it improves balance in which you must hold the body in one position. There is no transfer to active sports in which there is dynamic movement. To improve balance, especially dynamic balance which is needed in sports, you must do active movements and at the same time, keep your body in balance.
In addition, research has shown that if you maintain balance while executing a particular sports skill, you must maintain the same movement rate and skill technique. For most athletes such balance is usually maintained for up to 30 to 40 minutes without any changes in movement rate or technique.
However, with increasing fatigue, the athlete can still maintain balance, but there will be an increase in movement rate because he is usually unable to maintain or apply the same amount of force. As force production goes down, speed goes down. But, by increasing movement rate, speed can still be maintained and the athlete will still be in balance.
With even more fatigue, in order to maintain balance there is an increase in movement rate and frequency, in addition to changes in technique. These changes take place because of an even greater reduction in force application. But the key point here is that the athlete can still remain in balance.
Thus, as it should now be clear, balance is very specific and must be developed in execution of the sports skill. This is usable, functional stability that the athlete must display in execution of his skills in gameplay or in practice.
Sports balance is very specific and is a combination of several factors. This is why doing core/stability training to improve static balance is of little to no value in sports. It may be some benefit in everyday activities but even here, it is minimal.
For more information on static and dynamic stability, read Build a Better Athlete