There are always inherent dangers in learning how to do any physical activity. Minor injuries, such as bumps and bruises can be very common when learning martial arts and with children there is no exception. In fact this is often viewed as just part of the learning process and can actually help in the physical and physiological strengthening of the child.
If the techniques used are all well controlled and the training is very disciplined and well supervised then the risk of serious injuries are rare and considerably less than many other popular sports. To help find out how safety conscious your martial arts instructor is just ask them if they have current first aid certification.
In terms of personal safety there is of course much more of a danger if your child does not learn martial arts.