If it is a real Mermaid, then they gotta use the right bait or they'll never catch her. What they gotta do is dress some good lookin' man up as a Merman and have him hang around on the beach. He should have a full beard and long hair, cos Mermen don't shave or get haircuts, accordin' to legend. If she is there, she will eventually come out of the water to investigate, and that's when photos can be taken to prove or disprove this thing once and fer all, plus they might even catch her if they plan it right.
Now, if she was Hillary Clinton, fer example, then you could use a half-open suitcase with what looked like a couple billion dollars in it to get her to come out of the water, but that's another matter. So never mind. Just get a good Merman impostor in place, some guys with strong nets in hiding near him, and that should do it.