The Torah says that we may not eat anything without fins and scales if it is among the "sea-life" or of the "beasts of the sea" (Vaykira 11:10) which the Ramban thinks refers specifically to sea creatures that have two legs like those of land-creatures. Assuming that he's talking about mermaids, he implies that they have two legs, instead of fins, and are therefore not kosher.

However, the mermaids that I've seen in Disney movies do not have legs, but do have scales and (I believe) fins. Raavad, though, in his commentary to Toras Kohanim (to the verse above) includes sirens (a.k.a. mermaids) in the list of non-kosher animals. As far as psak halacha, the Aruch Hashulchan (Y.D. 83:10) also states that mermaids are not kosher. His reasoning is that the Torah only permits fish that have fins and scales, and even if a mermaid were to have those features, she is not actually a fish.

As always, be sure to contact your local orthodox rabbi if this becomes a matter of practical concern.