All right assuming slightly more advanced tech then todays, there are really just two options.
1) Engineered plague and a clean up squad. Disease won't kill all of them, but as long as it kills 80-90% it will be relatively easy to send in submersibles to harpoon and blow up the rest. Keep the kill squads around for a generation or two and whatever fragments of the Merfolk survived will be too small a population to be really viable.
2) Colony drop. Take several years or decades to maneuver an asteroid to land in the ocean. The shockwave, earthquakes, and decimation of a lot of the sea life will kill many of them. As soon as it's safe for humans, send in the kill squads and have a few nuclear bombs ready to destroy any large cities and pockets that survive.
Since this is a big ocean poison won't work as a realistic option, you'd have to sterilize the sea basically and at that point an asteroid would be cheaper. Nuking and attacking each settlement could work, but too many could escape unless you can cover the entire ocean in drones to track their movements.
It's easier and more cost effective to take the time initially to get them in one big hit.
Also to make the clean up easier, setting up refugee camps where the survivors could get food and resources as the rebuild, would be a good way to get as much of the surviving population in a few places as possible. Once the majority are in these places of safety set off a nuke in each one. That way the kill squads won't have to hunt down nearly as many to finish off the race.