The official name for someone who has a fear clowns is coulrophobia. It is a much more common phobia than most people would think. Most professionals believe the root cause of having a fear of clowns is some early childhood traumatic event. Traumatic only in the sense it frightening to the child.
The actual incident itself may have been long forgotten, but can still cause anxiety or a phobia later in life. There are some people with a fear of clowns that do not know they have a problem until it manifests itself in their adult years.
The most popular theory is that people are afraid of clowns because you can not tell a clown's true emotions. Because of the painted smile and exaggerated eyes, some with painted tears, people can not tell if the clown is as happy as he sees or if they actually have another agenda. Children in particular gauge their safety zone with people by their facial expressions.
Of coarse the entertainment industry has not helped the situation. If you have a fear of clowns and read "It" by Stephen King it is a perfect example of the industry playing on people's fears and carry them to the extreme. Films and movies have been riddled with depicting clowns as villains for years.
Treatment options include medication, hypnosis, and various other kinds of therapies that are developed to help us face down your fear of clowns until we can manage the situation ourselves. This particular phobia is easiest to handle by just avoiding clowns altogether. Parents should follow this rule of thought specifically for children. Do not attempt to cure your child by forcing them to confront clowns.
If your child shows any discomfort when being exposed to a clown consult a professional to ensure you are not creating a long term phobia that will be with them all their lives.