I think it could work with a minor tweak. If you stick the curtain to the outside of the window and keep it wet (including the surface of the window), it will cool the glass from the outside, which in turn will cool the air on the inside.
You will probably end up with condense on the inside (which would prove that it works) but compared to the original idea, that would be a minor problem. In fact it will lower the humidity inside making it more comfortable. In some countries it is quite common to have a small drainage under the window to dispose condense.
Here's a small example section drawing:
If you use a white cloth, it could keep the window more or less transparent (especially while wet) providing light while also partially reflecting some heating sunlight.
A dark curtain would not be a good idea. It may accelerate the rate of evaporation, but only because it becomes hotter, so that defeats the cause.
I guess some experimenting is worthwhile to find the ideal thickness and material for the cloth. If this proves to work out well, it may be interesting to develop an adaptive irrigation system specifically for this cause.
Update: Experiment results
I finally did some experiments last summer (2nd of August, 2018).
The humidity of the air on that day was 65%
My setup was with a table, having two shallow drawers under two glass plates, I had a digital thermometer for each. (this was taken in the morning while the table was still standing in the shade of a neighbors tree).
The thermometers would differ between 0.1 and 0.2 degrees Celsius when having the sensors in the same place.
I moved the sensors to approximately the middle of each drawer under the glass. The top thermometer display is in the drawer on the right.
Then I added the sheet to the left (I chose the left because it would be the first to get sunlight when the tree-shadow moved far enough. I also had given the left drawer the thermometer that gave the highest temperature. Not wanting to Bias the results in favor of what I would like to see.
The start was a little unfair off course:
I kept the sheet wet by spraying it every now and again.
And the results were quite startling!
63.8 degrees Celsius! I never would expect it to get that hot! Any way, compared to 48.4, the difference is about 24%. Now I guess my experiment is not like a window, it's almost flat, lightly slanted in the direction of the sun, so in real life with a vertical window I guess the difference would be a little less.
Later I did the same experiment with a dry sheet. The results were:
The difference, 49.4 vs 61.1 is about 19%, so the water is only contributing about 5% compared to the dry cloth. Note that the air humidity was 65%, so this 5% may be more if you live in a dryer climate, but less if the humidity is higher.
Sorry for taking so long to do the experiment and then the delay to report the results... better late than never I guess.