Which is the safest way to put on contacts on an airplane?

Carefully...: To put on contacts while sitting on a plane, use the lavatory first to clean your hands properly. Wait until the seat belt sign is off, since that usually indicates more stable flight which will allow you to handle the lenses with care. You should also have available rewetting drops, if possible, since many people find that their eyes get more dry while wearing contacts on a plane.

Not a problem: Airplane atmosphere is frequently drying so putting in a drop of lubricating fluid prior to inserting the lenses is reasonable. Otherwise the technique is the same as one the ground.