Nice button!
Best to let the dirt dry on the button, and then carefully use a toothpick to gently remove the dirt from around the details. Water can really be a hazard on some fragile brass and copper items, as the patina can be lifted right off along with details, leaving a rough brown surface. You've already taken the correct first steps, in not washing the button, as well as asking advice.
A light rub over the surface of the button after loosening the dry dirt, using a fingertip very gently, can highlight the details. A finish touch of a little mineral oil or pure Vasaline on the fingertip very lightly touched over the details, can also highlight the button. Leaving some lighter colored dirt in the recesses, can sometimes help define the details as well. Every button is a different scenario though, depending on the particular metal composition, soil conditions, and other factors. Also, buttons with gold gilt, and silver plate can benefit with some other cleaning methods, to bring out more of the original luster, if still present. Remember with most cleaning methods of relics, less is much better than overdoing.
BuckleBoy has also posted some great information here in the past, regarding the cleaning of buttons.
CC Hunter