Not sure what prompted your question, but to be clear, these devices are NOT "swamp coolers", which work well in dry environments using the effect of water vaporization to reduce air temperature. A byproduct of such coolers (in a closed environment) is increased humidity (and thus reduced effectiveness over time as the air becomes saturated).
The ice chest coolers do not work that way. They have a working fluid (water) in a closed loop that carries heat from the air (when the fan blows warm air over the heat exchanger filled with cool water) and dumps it into the ice (which melts and produces the cool water for the loop).
As wabower said, for practical purposes it's essentially a sealed system and all the water involved stays inside the ice chest and the heat exchanger loop. (What little vapor leaks out will not make a noticeable difference in cabin humidity levels.)