Hand trembling can be caused by reactions to natural body phenomenon, and because of certain diseases. In your case, your hands probably tremble when you are scared because of the sympathetic nervous system. The sympathetic nervous system is the part of the body that controls your ability to run and react quickly when danger occurs. It is the "adrenaline rush" that we feel during this time. In some people, this adrenaline rush when we get scared causes hand trembling. It is worse in some people than others, but usually does not have any significance to your health.
Other causes of a tremor in the hands include benign essential tremor, Parkinson's disease, and damage to a part of the brain called the cerebellum. All of these cause of a hand tremor occur all the time, not just during times of fear. The only one that occurs routinely in people your age is benign essential tremor. This is a hereditary tremor that occurs at rest and in general does not cause many problems. Your symptoms don't sound like this.
I suggest that you schedule an appointment with your primary care physician
. He or she can take a more detailed history of your symptoms and perform a thorough physical exam. This way, you can be sure that everything is okay. Good luck.