What would happen if I installed distributed water fountains in all of the ocean's dead zones?
That sounds like a method to attempt to increase the global humidity level.  I'm not sure how many areas we're talking, or who large they are, but &quot;all of&quot; just sounds like a lot.  It's at least going to be a massive humidifier the way you've designed it.
I would think if you want to achieve your goal without radically humidifying the air, it would probably work better to design your apparatus after aquarium aerators.
Somebody is studying this idea in the Baltic Sea. (thank you trond hansen).

A main threat for the Baltic is at first the human induced pollution through industries, agriculture, deforestation, private homes, shipping and tourism. Waste or erosion water due to these practices, contains lots of N and P that stimulate primary production. Also rising sea water temperatures due to global climate change make the oxygen availability even worse, since less O2 can be dissolved in warmer water (USGS, 2016). Moreover, also pharmaceutics or other artificial chemical substances that end up in waste water contribute for a great deal to the dead zones. Not directly by inducing primary production, but maybe even worse; the toxins they contain can either directly kill marine life or add up in the food chain and cause health troubles. Also reproductive success can be affected by certain chemicals.
In an attempt to stop the eutrophication and bring back life into dead zones, Swedish scientist came up with the idea of engineered re-oxygenation of the Baltic’s ‘dead zones’. Pumping relatively oxygen rich water from the surface into deeper anoxic layers, hereby breaking through the oxycline and allowing the water to mix again, must safe the Baltic according to the BOX project (Forth, 2015).
Facilitating nature to restore itself, instead of only focussing on banning the resources, could accelerate the battle against eutrophication in the Baltic. However, many scientists are sceptical about the idea. Prof. Conley (Lund University) mentions that there might be unforeseen consequences that could change the whole ecosystem. He also mentions that oxygenation could affect the reproductive success of some fish species. That’s indeed true, but Koster et al. (2005) found that increased O2 – levels actually improve the reproductive volume of at least cod, which thus makes it a benefit.
Apart from the costs of the BOX project, another point for consideration is the fact that treating the consequences instead of the sources might cause ignorance to obligations that try to reduce waste dumping, since it will be cleaned anyway.

In areas that are devoid of life due to a eutropic cycle (excessive nutrients in the water triggered a population explosion, which consumed all available oxygen, so everything suffocated), it would probably be helpful, so long as the aeration reaches the perimeter of the area.
However, if other pollution is involved, that changes things significantly.  Even if the initial death was due to eutrophication, developing life can be much more susceptible to toxic chemicals than mature life.  Because of this and other factors, what works for one part of the ocean may not work in other parts.
It's also worth noting that dead zones are not confined just to the oceans.  Something like this is being done in small lakes in Canada (Thank you Friddy).

Ecologically, many of these nutrient rich small lakes are classified as being in a eutrophic state. These lakes are typically shallow, with maximum depths of less than 50 ft and average depths of less than 15 ft. They are often landlocked or have only intermittent or seasonal stream connections. Because they are rich in nutrients, they are abundant in aquatic plants and invertebrates. Eutrophic lakes can be extremely productive and have the capability to grow trout and char very quickly and to large sizes. However, the same features that make accelerated growth possible in the summer can make it difficult for trout to survive over winter. Some of our most popular interior trout lakes are marginal in terms of their ability to support trout through the colder months due to a drop in dissolved oxygen levels Thousands of small trout lakes in BC’s interior freeze over for 4 to 6 months each winter. During that ice-covered period, the aquatic plants that grew through the open water seasons die off from lack of sunlight. Dissolved oxygen in the water is used up as this plant matter decomposes. As winter progresses, the volume of oxygenated water decreases to levels that become too low to support trout life, which results in a winterkill.
There are also small lakes located in the coastal regions that suffer from severe oxygen depletion during the summer months. This affects the ability of trout to survive as well as having impacts on the quality of the water for domestic use and other forms of aquatic recreation.

Their situation is quite different.  Rather than trying to resolve issues with pollution, they're using it to allow fish in ice-covered ponds to survive the winter.  There's also some lakes with issues during the summer the article doesn't go into detail about, which could be pollution or it could be natural eutrophication.  For the winter use case, it apparently works at two levels, as not only does it provide direct oxygenation of the water, but it also adds resistance to icing over directly above where the aeration is done, providing some surface that continues to be able to get oxygen from the air.
Both linked articles explicitly state this is just one tool to solve one particular issue, not a solution for all pollution, fish survival, and water quality issues.