Uh, Kelp, Crabs and Abalone are all seawater stuff. Type of fish would depend on market area, local water temps, etc.
Deep Blue has cages off Hawaii, last I heard, among others. However, it's probably more about how you go about getting approval. I would say, the initial setup would need to be set up as a study, with a university affiliation, along with the EPA, FDA, Federal and State Game & Fish/Parks & Wildlife, and whoever else should be involved. Part of the goal is to do so openly, mitigate environmental issues and make a working system, to produce commercial types of fish. However, Making it into a viable 'something' that has the potential to produce income, from a unique experience, is the Great American Pass-time.
Nothing of note, so far, but wikipedia (I know, but it's generally reliable)
Moving aquaculture offshore into the exclusive economic zone (EEZ) can cause complications with regulations. In the United States, regulatory control of the coastal states generally extends to 3 nm, while federal waters (or EEZ) extend to 200 nm offshore. Therefore, offshore aquaculture can be sited outside the reach of state law but within federal jurisdiction. As of 2010, "all commercial aquaculture facilities have been sited in nearshore waters under state or territorial jurisdiction." However, "unclear regulatory processes" and "technical uncertainties related to working in offshore areas" have hindered progress. The five offshore research projects and commercial operations in the US – in New Hampshire, Puerto Rico, Hawaii and California – are all in federal waters. In June 2011, the National Sustainable Offshore Aquaculture Act of 2011 was introduced to the House of Representatives "to establish a regulatory system and research program for sustainable offshore aquaculture in the United States exclusive economic zone".
By 2005, offshore aquaculture was present in 25 countries, both as experimental and commercial farms. Market demand means that the most offshore farming efforts are directed towards raising finfish. Two commercial operations in the US, and a third in the Bahamas are using submersible cages to raise high-value carnivorous finfish, such as moi, cobia, and mutton snapper. Submersible cages are also being used in experimental systems for halibut, haddock, cod, and summer flounder in New Hampshire waters, and for amberjack, red drum, snapper, pompano, and cobia in the Gulf of Mexico.
The offshore aquaculture of shellfish grown in suspended culture systems, like scallops and mussels, is gaining ground. Suspended culture systems include methods where the shellfish are grown on a tethered rope or suspended from a floating raft in net containers. Mussels in particular can survive the high physical stress levels which occur in the volatile environments that occur in offshore waters. Finfish species must be feed regularly, but shellfish do not, which can reduce costs. The University of New Hampshire in the US has conducted research on the farming of blue mussels submerged in an open ocean environment. They have found that when farmed in less polluted waters offshore, the mussels develop more flesh with lighter shells.
NOAA pdf on the subject...