Pondering whether protests actually work is a lot like asking if battles work. Multiple battles compose a war, and protests are but one tool in campaigns for social or political change. For a battle or a protest to work, it has to have a positive impact on the larger movement. Both methods can bring about change, for better or for worse, but so much depends on strategy and circumstance. In the case of protest, it also comes down to the willingness of the intended audience to hear the message.
Consider 20th century protest in Germany. Before Adolf Hitler rose to power in the 1930s, the Nazis staged protests against Jews, as well as other minorities. The Nazis pushed a vision for Germany's future that found widespread appeal among an economically depressed population still reeling from defeat in World War I. These early protests were successful in that they gave the Nazis the political capital they needed to rise to power.
In 1935, the Nazis instituted the Nuremburg Race Laws, which made anti-Semitism law and disenfranchised the nation's Jewish population. A year later, a Jewish man by the name of Stefan Lux protested to draw attention to his people's plight. After mailing letters to major British Cabinet Minister Anthony Eden, as well as several newspapers, Lux walked into the League of Nations headquarters in Geneva and shot himself in the building's assembly room. In one of his letters, he stated that he could find no "other way to reach the hearts of men" [source: Gilbert]. Sadly, historians largely view his protest as a failure. The world didn't listen, persecution escalated into Holocaust and it would be close to a decade before Allied forces liberated the death camps.
In addition to illustrating how protests can fail and succeed, these examples also underline that protest is not noble in and of itself. It can push cruel agendas as well as fight them. It can empower tyrannical regimes, as well as provide advocates a means of speaking out against them.