Combine fish sauce, sugar, vinegar, lime juice, chile flakes, and garlic in a small bowl. Add persimmons and let sit 15 minutes, stirring occasionally. Discard marinade.
Split onion lengthwise. Cut into 2-in. lengths, then separate layers. For each skewer, spear a basil leaf followed by a persimmon wedge, onion piece, mint leaf, and cilantro leaf.
*Find in the Asian-foods aisle.
Make ahead: Up to 2 hours, chilled airtight.