Big Mick's advice is still the best. I have traveled back and forth from Baltimore/New York to Ireland and Great Britain for some forty years with an old Vega F-5 long neck banjo and have never had a problem.
This travel has also included one of the worlds largest bodhrans (Mick knows), a Martin K-2, and assorted autoharps. Part of the secret is getting a good case and the second is in packing the instrument in its case.

Example of long-neck: The vast majority of any string instrument damage comes from side to side jerking, and not fore and aft. Every long neck case, including Carltons, has a mjor flaw in that they only support the neck at one and (special-ordered) two places on the neck. And in many cases this support does not keep the head of the instrument from striking the side of the case. Thus when the case is dropped, most likely on its side bumpers, the neck get broken near the first support. Remember the neck has been built to take fore and aft shock, since that is the direction the stings pull.

You should be able to fill an instrument case with socks and underwear to insulate the neck from any sideways movement..

Look at your current case, and once you remove the instrument, the result is an I.Q. test. If it's a banjo, have the hooks and ring holders worn a smooth spot in the fuzzy felt on the side opposite the handle? Look at the top; do you see evidence of where the foam inserts and fuzz touch the strings and go through to touch the fingerboard? Is the first place of contact the (GASP!) bridge? If some of these things can be fixed using some form of insert, use hard felt covered with moleskin or an adhesive softer felt, and NOT SPONGE ANYTHING, EVER! (Sponges, including the natural ones, can mate with the varnish finish of your instrument and wrinkle it) If you must, put the sponge in an anklet sock and ball it in next to the neck. Also loosen the strings as they may end up transmitting shock to the nut or the bridge. Once you are done, try and move the head of your instrument side to side as well as from the bottom or top of the case. If you can; you are not done.

This advice is as true for other instruments as it is for banjos… Your current case will tell you what you must do; not only for the airport, but for normal traveling about town, (or fighting your way out of a bar…)

Good Luck!