This answer assumes algae production on land in ponds or similiar structures, the question was about farming algae from the sea - so this answer may not be that interesting.
It is probably possible, but may very well be unsustainable or uneconomical. This very critical article lists several constraints that are unique to algae biomass:
- Water - is obviously needed in great amounts. While seawater is abundant in some places it is not so in deserts. In open pond systems we can expect much evaporation. Because of the competition over soil, deserts would lend themselves, but this would necessitate long range water transport with the associated energy cost.
- Infrastructure costs - especially in the case of closed bioreactors, there's much embedded energy (and money!) in the infrastructure
- In places where there are renewable energy laws that subsidize or otherwise support renewable energies, they may not account for algae biomass
One issue I don't see as a problem is fertilizer - unless the algae are burnt, the leftovers of the downstream process, whether oil production, AD or some other fermentation, will contain all the major nutrients.
Edit to add - Two other issues, from the comparison with land-based biomass:
- There's a huge body of knowledge and established technologies and infrastructure regarding farming - wether for food or other production. Algae biomass will require new inventions and infrastructure.
- Likewise there are huge bodies of regulations, subsidies and so on for the production of land-based biomass. Algae biomass may not tie well into these frameworks - this may actually be a good thing if regulations or subsidies push towards less sustainable practices.