Let's look at the down side. Even if you take a supplement with as little as 50-75 mg of zinc a day, this amount may decrease the good HDL cholesterol in your body. It is proven that this HDL cholesterol does protect you from heart disease, so you don't want to lower it. A small amount in a multivitamin-mineral supplement is not harmful. As an aside, preventing common colds is easier than treating them.
Humans have a natural tendency to touch their nose and eyes frequently during the day.
Check yourself out. The virus then flows down the lacrimal duct at the inside corner of each eye. This duct carries excessive tears and, of course, the virus from the eye into the nose. The virus has found a home and you have the common cold. So wash your hands often, avoid those with colds, and do not touch your nose or eyes.