Your diagram actually depicts a cross flow, but can we work with it.
You left out the 38% humidity in the incoming air (into the cooling tower).
To start, you should assume a steady-state process.
The mass flowrate of water is actually not constant... because water transfers into the air.
I do not believe this to be true. At 100% humidity, you should be able to calculate the percentage of water in the air exit stream. This amount minus the original water content in the 38% humidity incoming air is the total water removed from the cooling tower.
Remember to balance your flows in and your flows out.