Combine ground beef, green onions, salt, garlic powder, pepper, and ginger in a bowl; mix well.
Preheat the air fryer to 350 degrees F (175 degrees C).
Place several of the wonton wrappers on a plate or a cutting board without overlapping. Pour some water into a small bowl next to the plate.
Form about 1 tablespoon of beef mixture into a ball and place on one side of each wrapper. Dip your fingertips in the bowl of water and swipe some water around the edges of the wonton wrappers. Fold the wontons in half so that they form a triangle around the beef mixture. Wet your fingers with additional water and pinch the edges of the wrapper together. Place folded wontons onto plates, making sure they don't overlap, while you assemble the remaining wontons.
Brush one side of the wontons lightly with sesame oil. Place a single layer of wontons into the air fryer basket, oiled side down. Brush the tops lightly with additional sesame oil.
Air fry for 4 minutes, then carefully flip the wontons with tongs. Cook on the second side until the wontons are crispy and the edges are golden brown, 3 to 4 more minutes. Transfer wontons to a paper towel-lined plate and let cool while you cook the rest of the wontons in batches.