Properly maintained evaporative coolers / evaporative air conditioners will not transmit the disease because they produce no aerosols. Evaporative coolers should not be confused with cooling towers or evaporative condensers, which involve different technology. Secondly, in a quality evaporative cooler, water is drained from the unit when it is not required, so there is no standing water to rise to ambient temperature.
But above all, as we have said, well designed evaporative coolers do not produce any aerosol, meaning that even if bacteria were present in the cooler, there is still no mechanism there to transfer them to humans.
Moreover, there are no recorded cases of an evaporative cooler causing Legionnaires’ disease and research indicates transmission of the bacteria is not possible by the evaporative method.