Most electronic devices are X-ray safe, so airport security shouldn't be an issue...
Anyway, small electronic boards like Arduino are pretty rugged in nature. As long as you don't
- Power it on while it's on a metal surface (or anywhere that might cause a short circuit)
- Bend/knock it too hard (watch out those surface mounted components!)
- Immerse it in water for prolonged period of time
The board should be able to just handle it fine. I wouldn't worry too much about static electricity as the parts are already mounted on the board and all the high impedance pins are already connected..