A charcoal filter alone will not sterilize the air in itself, as bacteria will breed quite easily on the charcoal surface. This is often a desired effect, such as in aquaculture, as the aerobic bacteria that colonizes on the charcoal will break down ammonia produced by fish waste into nitrite and ultimately into nitrate. The charcoal will only serve to absorb contaminates out of the air and will allow fungal spores or bacteria to pass easily through.
A HEPA filter or micron filter will catch most spores but not necessarily all of them, and not bacteria.
A UVC Sterilizer Lamp is a common option, available from hydroponic shops and water purification places. These bulbs sterilize the air or water that pass by them and will easily kill any bacteria or fungal spore. They do however, have the unfortunate effect of producing ozone, which may find it's way into the reactor and kill the substrate. If a charcoal filter is used after the UVC lamp, the ozone will be absorbed by the charcoal and %100 sterile air will pass through.
Another simpler option that I read from a brewing manual was having an air lock filled with bleach or alcohol. When suction is applied, air will be drawn through the liquid from one side of the air lock to the other, which will result in "sterile" air.
Just remember that there is a very distinct difference between contaminated air and sterile air, as contaminated air is often sterile and relatively clean air can still have microbes present. Charcoal only serves to decontaminate air and only certain contaminates at that.