Whisk first 5 ingredients in bowl. (Peanut sauce can be made 3 days ahead. Cover and chill. Rewhisk before serving.) Fill large bowl with warm water. Working in batches, soak 3 rice paper rounds in water until softened, about 2 minutes. Remove rounds from water and arrange in single layer on work surface. Place 1 basil leaf in center of each round. Place 2 mint leaves atop each basil leaf. Place 2 papaya strips, then 2 cucumber strips atop mint. Arrange 2 cilantro sprigs atop cucumber. Fold 1 edge of each round over filling. Fold in ends. Roll up rice paper rounds tightly, enclosing filling. Transfer to platter. Repeat with remaining rice paper rounds. Cover with moist paper towel, then plastic wrap; chill. (Can be made 8 hours ahead. Keep chilled.) Serve rolls with peanut sauce.
