As a Christian, I believe fear is an integral part of humanness. In the beginning, God created man and he had no reason to fear anything, not animals nor God. The first instance of fear coming in is when he first sinned, i.e. when Adam and Eve first ate the apple from the tree of wisdom. They were suddenly aware of their nakedness and feared being seen by God (they hid behind trees when they knew he was near). So fear is a consequence of sin.
I don't mean that old ladies are afraid of being mugged because they've sinned for example, but by sin being in the world, fear is inevitable - fear of crime, and a fear that comes from feeling distant from God and his protection (even if we aren't aware of it). However, if we become reconciled to God by accepting Jesus Christ as Lord and accepting the sin in our own lives, we can become assured of Gods blessings upon us and having no reason to fear anything, not even death. (although many Christians will still fear, particularly myself!)
After all, if God is for us, who can be against us?