I have some "guitar on airplane" experiences. Having spent my highschool, college and grad school years abroad, I had to lug my precious classical guitar around airports quite a bit.
Here are my personal "golden rules":
a) A sturdy hard case is essential.
Those soft leather bags are a no-no. Even if you bring it on as a carry-on, you'll have to either toss it up in the overhead compartment or the hanger closet. This means that your guitar would get knocked around by other people's items if not well protected.
I used to have an ultra heavy duty hard case. I thought it was safe to throw my guitar in with those huge luggages. Guess what? The freaking ultra heavy duty hard case came out of the conveyor belt with a shattered shell. Thankfully, the damage did not reach into my guitar.
Which brings me to the next point....
b) Always insist on bringing your guitar onboard as a carryon.
Cry, bitch, raise your voice, bargain, flirt... do whatever it takes.
Generally, try to be at the gate early. If stopped by the attendant upon boarding, explained that you have a very expensive and fragile instrument (so what if its not that expensive... not the first time you told a lie) and insist on bringing in as a handcarry.
I usually ask to have my guitar stored in the jacket hanger closet. Those work well since the passengers are not likely to go around opening it as frequently as the overhead compartment.
c) Loosen the tension on your guitar strings for the flight.
I've read this somewhere once. My personal conviction came when I opened my guitar case after one long flight to find that a couple of strings snapped.
d) Use a dampit (humidifier).
The cabin air recylcing system dries up the air. Which is why you are frequently advised to drink plenty of water and apply moisturizer while on a long plane trip.
Chances are your guitar will be affected by the dry climate. But that might depend on the guitar. Some are more tolerant of dry weather, others less. You'll have to figure it out whether your guitar reacts badly to damp or dry weather.
Still a dampit will be very useful in preserving it from possible damage. These things are small and convenient to use (just soak the damper in water, clip it onto the edges of the soundhole, and cover the soundhole with the provided cover).
I find that once these points are observed, you can usually rest easy about the safety of your guitar.