Many large commercial buildings already do something similar, but instead of blowing cool, humid air through the structure, they use evaporation to cool the water itself in a condenser loop (with large cooling towers moving outside air across the water as it cascades down within the towers). The condenser water then enters a heat exchanger where it cools another loop of water that flows to the cooling coils in blower units throughout the building (you don't want dirty water that's been in contact with outside air going through cooling coils, so they are isolated loops).
Bear in mind, as others have mentioned, this approach is humidity-limited and is therefore rarely seen outside of arid regions. And also note that humidity levels tend to be highest at night when the air is the coolest, which further limits the effectiveness of evaporative cooling at night. Once relative humidity reaches 100%, you get absolutely no cooling from evaporation.