Mesh hats -- also known as trucker hats -- feature a foam, canvas or cloth front with an open weave in the back. Mesh hats have been in production for decades. If you've owned one for any length of time, at some point you'll need to clean it. Regardless of the materials used in construction of the mesh hat, you'll need to tread lightly as you make it sparkling clean once more.
Dab a clean cloth in some water. Gently rub the inside of the hat to check for colorfastness. If nothing comes off on your cloth, you're probably fine to move onto cleaning the rest of the hat. If you do see color, it's probably best to avoid cleaning it for risk of damage.
Spray problem areas like the inner band with laundry pre-treating spray. Allow this to set in for a few seconds.
Brush the pre-treated areas with a toothbrush. Don't use too much force, since this may deform any fabric on the front.
Rub away the grime and pre-treating spray with a damp cloth. Continue until the hat looks better, and your cloth comes away clean.
Place the hat on a drying rack to air dry.