What is the term which can be used to describe a person who is suffering from many phobias? I searched about this on Google but found nothing useful.
The condition is called polyphobia:
An abnormal fear of many things; a condition marked by the presence of many phobias.
The person suffering from polyphobia is polyphobic:
Many phobic individuals have more than one phobia and hence are polyphobic.
A person suffering from a number of phobias.
While the other phrases are more clinically correct, if I were to discuss such a person in daily conversation, I might say that they are Neurotic, as in
The man was neurotic; Everything alarmed him to some degree. He spent most of his life trying to escape his fears.
'Fraidy-cat is more colloquial. It refers to a very fearful person, afraid of everything, easily startled.
http://idioms.thefreedictionary.com/fraidy+cat - also as suggested here, you might use coward.
Phobophobia can also be defined as the fear of phobias or fear of developing a phobia. Phobophobia is related to anxiety disorders and panic attacks directly linked to other types of phobias, such as agoraphobia.
This is the name for a person who suffers from many phobias, formed from the prefix for many (poly) + phobic =polyphobic
Agoraphobia covers a lot of different phobic problems.
Strictly it is 'fear of open spaces', but tends to be used to describe anxiety states where the sufferer perceives the environment around them to be threatening.