Heat olive oil in a large pot over low heat. Add the garlic and cook until golden, about 3 minutes. Stir in the basil and tomatoes.
Bring to a boil and add the salt; reduce heat to low and simmer, stirring occasionally, until sauce has thickened, about 40-45 minutes. Stir in the balsamic vinegar.
Thread 5-7 spaghetti strands through each slice of sausage.
In a large pot of boiling salted water, cook pasta according to package instructions; drain well.