There is a distinction to be made between the fear of death and the fear of dying. The act of dying is something unknown and unnatural. There are those who fear dying yet they do not fear death. There are only two basic reasons why a person would not fear death. The Bible speaks of physical death, spiritual death and eternal death.
All men die physically because they are the sons of Adam.
All men are born spiritually dead because of Adam's original sin. If physical death overtakes a man while he is still in the condition of spiritual death, he will experience eternal death which the Bible describes as eternal punishment, separated from the presence of God. The moment a person dies physically, the soul departs the body and goes either into the presence of God or into hell to await the final day of judgment. If a person is not prepared for the final day of judgment, they have every reason to fear death. A right and proper fear of death is a great motivation to make sure one is rightly related to God through Jesus Christ the only mediator.