Wash your hands like you're in the hospital, and by that I mean all. the. time. Wash them in the airport bathrooms, on the plane frequently, and when you get to your destination.
Because the combo of extra hand washing and dry air from travel can make your skin feel uncomfortably parched (speaking from my own experience), keep hand balms and creams in your travel bag to mitigate the dehydration and cracking. Opt for subtle scents on the off-chance someone has an allergy.