In a food processor, pulse the shrimp, ginger, scallions, soy sauce, and oil until a paste forms. Add the water chestnuts, cilantro, and basil (if using) and pulse briefly.
Place 34 wonton wrappers on a work surface lined with wax or parchment paper. Place 1 heaping teaspoon of the shrimp mixture in the center of each.
Fill a small bowl with the water. Use a pastry brush or fingertip to brush the water along the edges of a wrapper. Fold it over the filling to form a triangle, then press the edges to seal. Use in Shrimp Dumpling Soup.