* As a street name: In this case, "Broadway" is a proper noun, as it refers to a specific street in New York City.
* As a symbol of theater: When "Broadway" refers to the theatrical district in New York City or the industry associated with it, it can be considered a common noun. For example: "She dreams of being on Broadway."
So, the answer is both. It's a proper noun when referring to the street, and a common noun when referring to the theater scene.