How does air movement affect evaporation?
It increases it.
Water evaporates or sublimates into the air based on the energy available. As you move further away from the boiling point the amount of evaporation decreases. It further decreases when you increase the amount of water vapor in the air. The drier the air the more readily available it is to take water molecules. We can see and experience this when we look at the humidex. In the cases of the humidex, it feels warmer out than it is because the humidity of the air does not readily accept more water vapor, which means that your sweat will not evaporate as readily.
Now, if the wind is blowing and it moves the most humid air (the air directly above the liquid water), and replaces it with air that is less humid. Now the air will accept more water vapor, allowing the water to evaporate easier. This is why it feels cooler when you have a breeze on a hot summer day. It isn't in fact any cooler, it is just allowing your sweat to evaporate more freely.