Until just a couple of years ago, my son was a very, very anxious air traveler. At the best of times he dislikes sudden loud noises, especially if he doesn't know what is making the noise. Starting when he was about 3 or 4 years old, he would often become extremely agitated when we got on an airplane. Sometimes he would put up a huge fuss: clinging to me, crying and struggling in his seat. Occasionally he would even refuse to put on his seat belt, and while we were in the air, he would not let go of my arm or my hand.
These days, he's still worried when we fly but much more relaxed. He's not the anxiety-immune type of air traveler: that would be my daughter who is more interested in where her toys are, what's in the play-package she got from the stewardess, and what show she can pick from the screen in front of her. My son is still somewhat anxious, especially at take off and landing, but these days it's more mixed with excitement and enjoyment of the flight.
From my own experience, here are some tips on how to deal with a child who is scared of flying:
Talking to your child about the flight ahead of time, and telling them what will happen can help with anxiety. For some kids it's good if you go into some detail: tell them that you will board the plane, find your seat, sit down, and then put on your seat belts. Try to describe what happens at take off and landing too, so that these things don't come as a complete surprise. Reading books, or watching movies about airplanes can help with preparation too.
Read more about travel preparation for children here.
In the moment, on board, when your child is frightened and maybe crying or struggling to get out of their seat, do whatever you can to calm them down: hold them, hold their hand, sing a song, and keep reassuring them that everything is fine. For my son, the physical comfort of having someone hold him was the only thing that really helped.
Though it is not ideal, you might have to put your child in their seat and put their seat belt on even in a situation when they are still very scared. That's not pleasant, but sometimes you just end up having to do it. Do still try to comfort them by reaching over to hold them. Just make sure that their seat belt is on or the crew will be on your case: that is their job after all.
This works for some kids. When my daughter had some mild anxiety at landing once (mainly because she was over-tired at the time), a strategic lollipop took her mind off it. In my son's case, bribes pretty much never worked. Bring a treat you know they love (preferably something that won't melt, make a mess, or otherwise be a bother on the flight).
Looking at a new book or toy, looking out the window, talking about the place you're going to (whether you're headed back home or is setting off on a new trip), making silly faces, bringing a puppet to play goofy games with, playing peek-a-boo with a blanket... you get the idea. Just take their mind off what they're scared of.
If there are individual TV-screens on your flight, try to find a movie they like, bring a game they like, or an i-pod with their favorite music or story.
Talk about it
Try to pinpoint what is making your child fearful and talk about it (this doesn't always work for very young or very fearful children). If noises are causing the fear (as in my son's case), then try to explain what's making those sounds. Bring a book about airplanes and refer to it while you're flying. Make it all like a game: do you remember what's making that noise? What noise will we hear next? Do you know why the wing is changing shape?
Fear of the unknown can be very powerful, and when that unknown includes all the sounds, sights, smells, and physical sensations on board an airplane, it can be overpowering. Finding out why things get bumpy in the air sometimes, what's making that "clonk" sound, or why the airplane is all of a sudden tipping it's nose down can help both kids (and adults) feel safer.
I know parents who give their babies and toddlers Gravol before a flight to make them sleepy and keep them calm on board. I have never done this myself, and wouldn't personally feel comfortable doing so unless it was an extreme situation and my child was not just anxious but also sick.
If your child is frightened to the point of it becoming debilitating and you still need to fly, you can always speak to a doctor and get their opinion.
It's important to try to stay calm yourself even if your child is anything but. That's not always easy to do of course, especially if your child is freaking out and refusing to put their seat belt on, screaming, wailing, and kicking their feet
Most kids do calm down eventually, unless something is physically bothering them like an ear infection or other illness. Even when my son was extremely anxious about flying, the excitement of the trip would take over once we were in the air, even though he never totally relaxed. Take-off and landing are usually the worst times for anyone with a fear of flying. Any kind of turbulence can also be very frightening for a child, but then let's face it: nobody likes turbulence!
My personal experience is that lots of preparation, lots of practice (flights), and learning more about airplanes are the best ways to help reduce a child's fear of flying. Each child is different, so try to find what strategy works best for you, and then deal with the situation the best you can. Which is what we parents do pretty much every day, whether in the air or on the ground.