Heat the oil in wide-bottomed pot set over medium-high heat. Add the onions and prosciutto and cook for 3 to 4 minutes. Add the garlic and cook 1 minute more. Add the wine and bring to a simmer. Add the mussels, cover, and cook just until they open. Sprinkle with salt, pepper and the fresh basil. Divide the mussels and cooking liquid among bowls and serve.
Note: Discard any mussels that do not close when tapped before cooking, or do not open after cooking. Both are signs the mussel was dead prior to cooking and should not be eaten.