I developed a fear of flying back in the late 1990's. I love going to the airport to watch airplanes take off and land, I know that flying is the safest mode of transportation, I know that airplanes have backup systems, that they are built to handle serious turbulences, that airplane pilots are high trained professionals, etc... It doesn't matter.
For many years it had a serious impact on my life because I lived in the US while my family lived in France. I had to do the journey across the pond at least once a year. There was no alternative to flying so I just had to deal with it and it was not pleasant - to the point where I loathed going to visit my family.
When I lived on the east coast it was bad enough, but then I moved to California and the 12-hour flight was an ordeal. But again, there was no alternative. From California I first tried to break up the journey by flying through an east coast hub but I quickly realized that it added a number of complications (additional risk of delay, cancellation, lost luggage, etc...). So I ended up taking direct flights. On long haul flights, flying business class helped. A cocktail or two sipped very slowly did too. Sometimes. I haven't yet found the magic formula.
When I moved back to Europe, I thought that my days of flying were over, at last! But it wasn't the case. Last year I had to fly to the Canary Islands for a family vacation and in January I flew to Turkey. I could have driven to Turkey from here, and I did last year. But it was a long, long drove through Italy and southeastern Europe and, while I enjoyed it at the time, I must admit that the 4-hour flight was far more convenient.
Despite the sheer terror of boarding an airplane, I have never backed out or canceled. I keep a spreadsheet of my flights to remind me of all those small victories. Since developing my fear of flying I have managed to fly over 300,000 miles. Yes, "managed" is the appropriate word.
Now for my little flying story...
I was flying from San Francisco to Newark one year and I was sitted next to a man who seemed to be very at ease with flying. I never, ever talk to the person next to me when I fly but for some reason, on that day, I initiated a conversation with this stranger. I found out that he was an off-duty airline pilot. We exchanged pleasantries for a while. Halfway through the flight I started smelling a burnt smell (kinda like melted plastic) and I told him about it. He confirmed that he smelled it too and called a flight attendant. Shortly after he was called to the cockpit and when he came back he told me that it was serious and that we were diverting to Chicago. And yet panic didn't overtake me. In fact I was strangely calm until we landed safely in Chicago. And the next day I boarded a flight to Europe.
I had a similar experience. I was returning home from a week long business trip that was full of long days. On my flight, I had an aisle seat. There was a younger woman in the window seat, with no one in the middle seat. between. I gave a courtesy smile and brief hello when I sat down. she just had this pale stare, and nodded. No problem, I usually do not talk on flights and I intended to doze off. I remember the end of the safety announcements... next thing I feel is someone tapping me on my shoulder. It was the woman, starting at me with big eyes. "You slept through the takeoff!" she said. I just nodded and said yes, I had a long week. "How can you do that??" Well, so much for any nap on this flight. She wanted to talk, she was a very nervous flyer and but it made her less nervous because I was so relaxed. When we were about to land she asked if I minded her sitting in the middle seat to hang on to my arm. I said sure. She had quite the grip on my arm as we landed. But she was appreciative, saying "I wish when I flew I was always next to someone like you". My good deed for that day (probably for that month 
