Most people I have spoken to do not fear death as such; rather, they fear the process of dying or certain things that might be entailed by that process.
By this, I mean specifically the possibility that we might die before accomplishing all that we set out to achieve, or that we will leave our loved ones behind, or that they will not be able to get on with us, or, as you mentioned, that our consciousness–the core of our life/experience–will come to an end.
For some people, it might be a consolation to know that some form of our being will continue on after our body has ceased to function. The larger river of life, of which our human being was only a current, will continue to move on. Others will not be comforted by this fact; their attachment to their loved ones or their fear of no longer being aware of the world is too great for them. For them, this message will not hold much might in terms of emotional comfort.
The comment that life continues on after the body began to function just as it did before we were born is not meant to be a final answer to the human fear of death and dying. It is rather intended to point out that one conception of death–as annihilation–is false. Death is transformation, a process in which the body ceases to function and begins to break down into component parts that will recombine to take new forms.