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I was wondering that too; personally, I have no idea how to even begin answering the question. It would seem that at least some protests do work, as evidenced by the civil rights movement during the Martin Luther King era; but I don't know if this is true in general.
I think protests work if there is already a critical or near-critical mass of support in the relevant decision-making body (legislature, courts, civil service, etc.) Protests rarely change minds, but they can give already-sympathetic people a new impetus to take action in this area rather than another.
ETA: It also helps if the protesters have specific, focused demands, like "end segregation," or "bring the troops home."