Somehow the mighty Western navies see anti-piracy as beneath them, which is remarkable since in years past they collectively managed to handle outbreaks quite well. We often forget that the vaunted Golden Age of Piracy in the early 1700s was short lived when the Royal Navy got involved to stamp it out. Later the Barbary Pirates were easily suppressed after America and Britain finished fighting one another in the War of 1812.
I think if the US and her allies can humble themselves a while, they could easily end this scourge while it is mostly contained in the Western Indian Ocean. Taking the counter-insurgency lesson learned after much trial and error on land and applying it to the sea, the Navy might just find a renewed mission and favor with the public, which it has been desperately seeking since the Cold War.
Mike is right, the scourge can be ended. As sailors, we recognize the threat that pirates pose to freedom of communication on the seas, and know instinctively that they must be eliminated. It is not beneath a sailor to kill a pirate, and the problem is one entirely of national will.
The young American nation sent its Navy after the Barbary pirates because her vital national interests (i.e., the ability to freely trade in the Mediterranean and eastern Atlantic) were threatened. The Royal Navy motivation was the same, except that they made a global rather than local commitment.
The United States and her allies don't need to "humble themselves a while," they need to recognize that (a) pirates are a threat to global commerce wherever they operate, and (b) navies, not police and lawyers, are best suited to the task of eliminating global threats on the seas.
The real challenge to this will be convincing the American public that their national interests are threatened, because so little international shipping rides in American-flagged or American-owned hulls. According to the data on the Maritime Administration web site, only 1.47% of the total deadweight tons in the world fleet sail under the US flag on 286 ships; 4.26% of the world tonnage capacity is US-owned - 207 dry bulk, 52 container ships, 69 RO/RO, 318 tankers, and 38 general cargo ships. Given that most of the US-flagged fleet is likely limited to service between US ports, their risk of a pirate encounter is extremely low, and the American-owned ships in international trade are likely operated with foreign crews.
The clear and present danger to world freedom just doesn't register on the American consciousness yet. It will take a concerted effort by the national leadership to inform and persuade them. I'm afraid that the influential bandwidth needed for this effort just doesn't exist in Washington today, especially with an election looming.