The only two things mankind has profitably extracted from seawater is salt and magnesium, but I don't think we've really tried hard enough to give up on extracting the more interesting things, we just need to combine the processes a bit better.
The first step to profitabity in a ocean mineral extraction
scheme will be to sell the water. Assuming distilled water is marketable, the rest of the scheme gets easy, due to a certain material: PVC. Uniodized common NaCl salt is so easy to produce that the market for it will saturated, but that of course will be sold too, likely at a "loss", for whatever price the market will bear.
PVC is a very simple plastic, to make it, one needs ethylene, or c2h4, and hydrochloric acid. It is considered safe for water pipes. It's is dangerous if it burns, but hey, most things are dangerous when they're on fire.
Ethylene can be produced via hydrocarbons, or via plants. Since it's chemically only carbon and hydrogen, there should be no problem developing a cheap process for ethylene production on the sea.
Hydrochloric acid should be equally easy to produce, especially as a waste product of magnesium production.
the PVC thus made could be molded into small seaworthy craft and storage containers. Some can be filled with refined common salt and stored whereever, but I imagine that most of the salt will be excess salt and have to be re-mixed back into the ocean, hopefully in a eco-friendly way, and far away from the mineral extractors.
The remaining barrels can be used to store various mineral almagams, as more and more minerals are removed from the mix.
At first, the calcium and potassium, et all will be too abundant, but once you get past those, then you get at the rarer metals. You'll have sold astronomical amounts of water by the time you get much gold out of it, but the magnesium is of course useful, and large amounts of PVC created will be a carbon sink, as well as a fuel source which can be cannabalized in times of fuel crisis.
The energy source used for such operations is matter of other discussion, but there will be an obvious solar element to the endeavor, which will likely be supplemented with whatever else is appropriate.
There might be a wide variety of ships that would need to be floated, the "standard" ship which produces salt, distilled water, and a brine, then a ship which extracts the magnesium and calcium from the brine, then another ship to manipulate the next "waste" stream to extracted the next profitable mineral.
These ships could topographically be arranged throughout the ocean to minimize transport, avoiding hurricanes or other hazards when necessary, the pvc ships and containers stacked up and tied down when necessary.