Combine oil, vinegar, basil and garlic in a bowl. Add the shrimp and marinate overnight in the refrigerator. Preheat broiler.
Drain shrimp. Cut each piece of prosciutto in half, lengthwise and wrap around each shrimp, securing well with a toothpick.
Saute the shrimp in a non-stick pan, that has been heated and lightly brushed with olive oil. Cook only to seal the prosciutto to shrimp. Place each piece on broiler pan and grill under broiler for 2 minutes on each side. Careful not to burn the prosciutto or overcook the shrimp.