The really short answer: If the wet-bulb temperature in your basement is sufficiently low, then yes. Otherwise no.
Note that for a small poorly-ventilated basement, the humidity will rise from respiration, thus increasing the wet-bulb temperature. For a well-ventilated basement, the air temperature will approach the outdoor temperature, thus increasing the wet-bulb temperature.
If you have a giant, well-shaded, pool of water, the water temperature will usually be below the peak air temperature (water has high thermal mass, so it will take a long time for it to come up to the ambient temperature, even at high humidity, where evaporation effects are negligible) so that's another non-AC method of staying cool.
In any event pools of water and basements don't scale to densely populated areas.
Note that for a small poorly-ventilated basement, the humidity will rise from respiration, thus increasing the wet-bulb temperature. For a well-ventilated basement, the air temperature will approach the outdoor temperature, thus increasing the wet-bulb temperature.
If you have a giant, well-shaded, pool of water, the water temperature will usually be below the peak air temperature (water has high thermal mass, so it will take a long time for it to come up to the ambient temperature, even at high humidity, where evaporation effects are negligible) so that's another non-AC method of staying cool.
In any event pools of water and basements don't scale to densely populated areas.