For many people, they don't like the loss of control that they feel when flying. Lack of knowledge about how an airplane works often compounds this along with anxiety created by the manner in which the press covers airplane crashes. It might help to allow the nervous flyer to sit in the cockpit sometime before the flight, explain to them the various instruments and controls and give them a walkaround of the plane taking care to explain how airplanes fly.
I also point out that general aviation flight can and usually is just as safe as airline flying. A properly maintained private airplane with a properly trained pilot is mechanically just as safe as an airliner if not safer. Just because an airplane is old doesn't necessecarily mean that it is less safe compared to a newer airplane. Having grown up around planes, I realize that many of the GA
crashes are due to either lack of pilot experiance, poor maintenance or a combination thereof. Furthermore, I will go anywhere on any airplane that is reasonably maintained.
"When a man runs on railroads over half of his lifetime he is fit for nothing else-and at times he don't know that."