For the outdoor grower, there is little need to consider air flow, beyond making sure that your growing area is not too windy. However, air flow at the leaf surface is an import consideration for the indoor grower. Can you grow successfully without air circulation? Sure! I have grown in a small closet without any fans at all, but it seems like the plants do better with a small fan blowing in my grow area.
Plants takes in carbon dioxide (CO2) from the air, and emit oxygen (O2) as a product of their metabolic processes. Plants also transpire, which involves the evaporation of moisture from pores in the leaves. As this moisture is leaving through the leaves, capillary action tends to ‘pull’ new moisture and nutrients into the plant through the roots. Slight air movement at the leaf surface can be helpful for these processes.
I recommend using a small oscillating fan in your grow area. I have one of these that I set on ‘Low’ aimed slightly above my plants, and I only run it during the day. You could run it 24 hours a day, but I have not found it necessary. Shoot for a level of air movement in your room that gets the leaves fluttering slightly. You don’t want a strong fan blowing right on the plants, this can dry them out.
On the subject of ventilation, I have found that when you grow in a small unventilated room, such as a closet, the transpiration from the leaves tends to make the room feel humid, which could potentially lead to mold issues. If you are using any sort of artificial light that emits heat, the room can also get a bit warm. High humidity will inhibit further transpiration, and high temperatures are not ideal for some cool-weather plants, like lettuce and cabbage. Consider some sort of ventilation if your room is small. You don’t need to install ductwork, just open up the door from time to time or crack open a window to let the moisture escape.