Bring a large pot of salted water to a boil. Combine 1 tablespoon olive oil and the breadcrumbs in a small skillet. Cook over medium heat, stirring, until the breadcrumbs are golden brown, 6 to 8 minutes. Season with salt and transfer to a small bowl.
Add the spaghetti to the boiling water and cook as the label directs for al dente. Meanwhile, heat 3 tablespoons olive oil and the garlic in a separate large pot over medium-high heat. Cook, stirring, until golden, about 2 minutes. Stir in the chiles. Add the mussels and wine and bring to a boil. Cover and cook until the mussels open, about 5 minutes (discard any unopened mussels). Drain the mussels in a colander set over a large bowl. Pour the liquid back into the pot. Cover the mussels to keep warm.
Using tongs, transfer the spaghetti to the pot with the mussel liquid. Cook over medium-high heat, tossing, until coated. Add some of the pasta cooking water, 1/4 cup at a time, if the pasta seems dry. Stir in the parsley and season with salt.
Divide the pasta among bowls. Top with the mussels, breadcrumbs and basil; drizzle with the remaining 2 tablespoons olive oil.