Use a sharp knife:
A sharp knife simply slices through the tissues and does not cause the chemicals to disperse much. A dull knife messes up the cells and more chemicals are released.
Before you plan to cut onions for use, keep them in the freezer for a quarter of an hour. The freezing reduces the amount of chemicals released into the air when you cut the onions. However, do not freeze them for long because it will become harder for you to cut and the onions also become mushy.
The above two tips were to reduce the chemicals released. This one is to keep it from reaching your eye. The best protection is to wear glasses before setting to work. Contact lenses are also alternatives. Cut the onion at an angle away from your eye so that the chemicals do not enter you directly. Cut in a way that the tube is away from you because the flesh is tubular in nature. Working under a fan also helps disperse the chemicals.
Chewing gum is one of the famous methods with little scientific backing but a lot of followers. Chew gum while cutting the onion. Another method is to light a candle near the cutting board. The flame will attract the chemical gases.
A wet onion releases very little chemicals. You can cut the onion under water. Cutting near a water source also helps because the water draws the chemicals. Soaking it up and then cutting it also achieves the same result.
Do not skin off the onion altogether. Work in small portions and take off the skin bit by bit. This way you will not expose the whole surface altogether and the chemical release will be minimal.