My ghetto/redneck solution is a towel with the end dipped in a bucket or tub or sink hanging an inch or two from the intake of the fan. As the water evaporates (and cools) more is drawn up into the towel via the wick effect. Definitely not as fabulous as an elaborate system of ropes and pulleys seen in those fine examples of ingenuity, but it works and allows me to be very lazy.
The cooling effect of evaporation doesn't affect the vapors, infact it is just the opposite, it only cools the body from which they are evaporating. All you are doing is making your area more humid, thus making your sweat less effective.
However, I just stuck my hand in front of the fan, and I believe my experimental method is sound. If you have trouble reproducing my results, let me know. I'll have my landlord fix the physics in my house and try again. (Or, you could add the cooler temperature of the water in the basin to the calculations in your model.)
Actually, the evaporation effect will reduce the temperature of whatever is around it, in this case the air. I don't know whether the effect is dramatic enough to provide significant cooling for an entire apartment, but it's probably enough to make you feel it. Do a calculation using the heat capacity of water and referencing it against how often you have to refill the bucket.
Evaporation is what separates nice, cuddly good-weather clouds from thunderstorms.
