I agree with Jim's comment above. As the graph here suggests, the supply of wild fish appears to have been flat since the 90s, and the increase in demand has been met by the supply of farmed fish. So I think it's likely that consumption of wild fish will just cause someone else to consume farmed fish instead.

With regard to fish oil: Most of it originates from small wild fish such as anchovies. There's an entire industry dedicated to harvesting fish oil and fishmeal, and most of it is used as feed for carnivorous farmed fish like salmon. Fish oil seems to be mostly supply constrained as well, and the aquaculture industry is responding by feeding carnivorous fish more plant oils. I've written about this here and here, and should probably move these to the EA Forum now that less polished posts are encouraged.

Brian Tomasik has written about this here:

Substitution across fish types

The analysis becomes more complex when considering other wild fish species and fish farming. A decrease in the wild population of a given fish species may not reduce total fish consumption but might merely shift consumption to other fish.

Fishing down the food web

Big fish tend to be overexploited first, since they're easier to catch and take longer to mature. As populations of big fish decline, the result is fishing down the food web, i.e., harvesting smaller fish species. To produce the same amount of fish meat, more total fish will need to be harvested. Even if Fred weighs the badness of killing fish by the brain complexity of the fish killed, as long as Fred's brain-complexity function is less than linear in the mass of the fish being caught, then catching a given mass of smaller fish will be worse in Fred's eyes.c Assuming the total mass of fish harvested stays roughly the same now and into the long run, then eating more big fish and thereby reducing their long-term yields will have made things worse for Fred.

Fish farming

In addition to causing consumption of more small wild fish, overfishing of big species in the wild creates pressure to produce farmed fish. Some farmed fish are fed smaller wild-caught fish, which means fish farming can significantly increase total fishing. This would be prima facie bad in Fred's eyes.

However, there might be cases where fishing pressure on smaller wild-caught fish who are fed to bigger fish is sufficiently intense that populations of the smaller fish also decline. One example of this was the Peruvian anchovy fishery, which had supplied fish meal for livestock but which collapsed, leading to increased demand for soy protein instead.

Unfortunately, other fish-feed sources might be even worse than wild-caught fish, such as insects. Farming insects to feed farmed fish would significantly multiply the number of animals killed by humans. That said, fish feed may also include plant-based ingredients.

How much would substitution happen?

In rich countries, I would expect that demand for fish would remain pretty stable in the face of further fishery declines. As there become fewer big wild fish, prices of fish increase slightly, which slightly reduces the quantity demanded. And some eco-conscious consumers might cut back on fish in response to overfishing. But on the whole I expect the effect to be modest for affluent omnivores.

However, the effect might be more pronounced in poor countries, which may lack the resources to undertake aquaculture. According to the UN FAO:

In many areas of sub-Saharan Africa and South Asia, [...] fish consumption levels remain too low and they are failing to benefit from the contributions that fisheries and aquaculture are increasingly making elsewhere in terms of sustainable food security and income.

Given that up to 90% of US seafood comes from outside the US and that most rich countries import a lot of seafood from developing countries, increased consumption of fish in rich countries might indeed reduce long-term fishing by poor countries (to the detrmiment of indigent people in those countries).

Total fish production

One macroscopic perspective from which to assess the total impact of fishing is the following graph (compiled by Earth Policy Institute). [Graph here] I think the trends in the graph might be overstated if these figures include China, which is a major fish producer that's widely believed to overreport its fish numbers.d So the increase in fishing is possibly less dramatic than what's shown. But assuming the overall trend of wild + farmed fish is still an increasing one, then this is a bad sign from Fred's perspective, even if Fred only cares about fish in proportion to their mass (so that Fred wants to minimize the total biomass of fish caught). This is weak evidence that further demand for fish would make things worse for Fred, since most of the trend in the graph was probably driven by growing fish demand.

Of course, some fishery doomsayers might claim that the upward trend in the graph can't last forever (especially since fish farming often relies on wild-caught fish), and if they're correct, then drawing conclusions from the portion of the curve we can see now would give the wrong impression. But given that some fisheries are recovering, especially in the US, I'm personally skeptical about "end of fish" scenarios.

Conclusion

When considering substitution of fish consumers away from a declining fish species toward other species and fish farming, the effect of marginal big-fish consumption in the short run on total long-term fish harvesting becomes unclear. It's not obvious how much a reduction in populations of big wild fish reduces total fish consumption especially by the global poor (which could be good from Fred's perspective) versus how much it merely shifts consumption to other fish types (which could be quite bad from Fred's perspective).

Note that this is also relevant for the question of whether eating more wild-caught fish is good for fish. If humans continuously restock waters with hatchery-produced juveniles to compensate for wild-caught fish, fishing might not affect wild fish populations in the same way.