By default a pirate is only intimidating to a civilian. Anyone with actual combat prowess can beat a pirate easy. A pirate just picks up a sword of gun, yells "RAAAUGH!", and people cower because he's got a weapon and they don't. Also, the stench of cabin fever. Do you know how filthy those boats become? As such, a sober pirate is doing nothing more than trying to aim an already inaccurate pistol or swing a sword without any proper training or skill level.
Ninjas, on the other hand, are not only trained in combat, they're pretty much Spec-Ops. They are the sort of warriors that go in when you want everyone to believe Death himself strolled through, leading souls to the netherworld. Make one of them drunk, and you just remove his prior inhibitions. Of course, if you make him shit-faced, then the pirate may have an advantage. But otherwise, the ninja still possesses all of his skills, but loses any restraint. That is one dead pirate.
Now, some people may be tempted to bring up specific characters, such as the Dread Pirate Roberts, Captain Jack Sparrow (oh yeah, he'd definitely lose) or various characters from One Piece. Ninja fans can easily counter with Dr. McNinja, the Teenage Mutant Ninja Turtles, Ryu Hayabusa, etc. When specific characters are listed, it no longer becomes a matter of "Pirate vs. Ninja". It becomes a very specific fight with very specific attributes.