Apparently the world's large ocean fish are disappearing. The Volokh Conspiracy's "Juan non-Volokh" argues for fisheries management as the solution. I'm skeptical, given the enormous economic pressures facing fishermen in these situations. After all, as stocks deplete, the value of each catch actually rises, often faster than the extra costs of finding and catching the now-rarer fish.
The best hope, I think, lies with fish farming. Here in the Pacific Northwest, salmon farms are seriously undercutting the wild salmon fishing industry, by providing such an abundance of cheap high-quality salmon that only complete fish snobs (of which there is, alas, no shortage in these parts) would bother to pay twice the price for the wild-caught stuff.
I have no idea whether deep-sea fish can be farmed, but in any event the availability of a wide variety of farmed fish would inevitably cut demand for undomesticable species, in the same way that wild pheasant is rather less in demand than it would be in the absence of domesticated chicken and turkey. And farmed fish are not only likely to be substantially cheaper than wild varieties, but they can also be bred and selected for better taste, nutritional superiority, and low levels of contaminants (whereas large ocean fish are said to be worrisomely contaminated with heavy metals such as mercury).
It's certainly possible that some species of deep-sea fish will still be in high demand as a delicacy, just as the existence of farmed mushrooms has not eliminated the market for truffles. But they would be the exception; the tuna that gets packed into cans, for example, could easily be replaced with cheap farmed fish--especially as the former gets rarer, and thus more expensive, to catch.