The taming of the Plesiosaur is by far the most difficult in the game. Generally, the Plesiosaur's level seems to create an extreme difference in the resources required to tame. Due to their very fast torpor decrease (about 8-10 per sec), you may need up to 1000 narcotics to maintain a higher level Plesiosaurs (30-50) unconscious, however the lower levels may require less.
Bows will not work underwater and wild Plesiosaurs will not swim to the surface of the ocean, so you will need to either use a crossbow or lure the Plesiosaur into a cave where they can surface.
Another technique is to use the crossbow and attack the Plesiosaur from above. They seem to lose interest in you when you're near the surface, however, and this allows you to take shots at them while still moving far enough away to make them lose interest. Repeat this process and you should be able to knock it out without much issue. This still leaves the risk of Megalodons and other Plesiosaurs attacking you and your tame, though, so be cautious.