When a water balloon is thrown towards a person it bursts when an attempt is made to catch it because the material that the balloon is made of cannot withstand a large force. The water filled in the balloon exerts a large pressure on the surface. Due to this a small force is enough to break it.
To catch a balloon that is moving, the momentum of the balloon has to be reduced to zero. It is also essential that this be done as slowly as possible in a manner that does not result in large force acting on the balloon's surface. To accomplish this when you catch the balloon continue to move your hand backwards slowly reducing the speed to bring them to a stop over a large period of time. In addition to this use the maximum surface area of your hands to catch the balloon so that the force is distributed over a larger surface of the balloon's skin.