Here is a list of sea things edible to humans and human efforts at cultivating them: https://en.wikipedia.org/wiki/List_of_types_of_seafood
If mermaids can digest other foods in addition to these, they will have a larger range of diet. In addition to Andrei's fine points, we do have some basis for agriculture. The sea cucumbers, the kelp, the seaweed, and even edible microphytes. Some of these are a good source of nutrients, thankfully, which in addition to fish and other things should make for a stable diet.
For cultivating fish, crabs, etc., you will need a few things:
1) Environment, they can't freeze or the like. This can vary a bit with season, depending on the depth on the ocean.
2) Food source, you can't have your food starve or you'll starve. The farm's limit is its food's limit. If you can cultivate your farm's food somehow (I am no expert on how), you will cultivate your food. Note that with migratory fish you rely on seasonally, they have the advantage that they get their food source elsewhere.
3) Control, you need to be able to control this farm to an extent, or else it just becomes hunting. If you could enclose your food into a small environment they're easy to catch from, that works--but you have to keep in mind the possibility of them running out of food in this small environment, or spreading disease if they're too enclosed. Clam and kelp gardens are simple, along with other vegetables. Fish are trickier, most will need quite a bit of space, and good places to lay their eggs, and their full diet.
There are various ways you can control this. If you know when the salmon come in and how they get to their grounds, you could set up traps that allow you to capture a percentage of the salmon, and let the rest through to breed. If your mermaids can stand fresh water for a while, they could also wait till most of the salmon have spawned, then swim in to harvest the adults before they die naturally (and some caviar, if they're wasteful). I'm not sure what hunting of post-parent salmon is like.
You might even be able to breed the fish to the point where they'll start to have instincts that makes farming them easier. Other than that, dolphins and whales would be good tamable animals for work, defence, and possibly food. Seals might be, as well.
Sadly, I don't know enough about the needs of fish, to know how to set up these farms, how much space they need, what places for eggs, how much food and how well you can cultivate it. Real examples of fish farming will probably give a good idea of this.