Biological products will need to be cheaper & safer to compete with the current favorite, BTI
If you want to purchase a natural control for mosquitoes, you'll probably be sold something with Bti in it. The market is currently dominated by products containing Bacillus thuringiensis israelensis or Bti, a bacterium that targets mosquito larvae. It's relatively easy to produce and ship, and is effective at killing larvae when placed in the water where they breed. Any future natural, biological controls will need to be better than Bti, before companies can afford to spend significant research money to develop them.
But it's not perfect. The main problem is that it's useless against mosquitoes that breed in a nearby swamp, a neighbor's pond, a salt marsh, tiny hidden puddles, or other areas that are too large, impractical, inaccessible, or otherwise impossible to treat. Since Bti kills only larvae in the water, once mosquitoes are adults and flying into your backyard, you'll need to find other ways to get rid of them, such as traps or sprays.
In the future, additional natural predators and enemies may be developed, if practical problems can be overcome. Here's a look at what may show up next, in the biological warfare against mosquitoes.
Copepods are tiny crustaceans, less than a couple millimeters long and barely visible to the naked eye. They are natural enemies of the first and second instar (the smallest sizes) of mosquito larvae. The copepods attack and eat larvae, but can also survive on other food sources in water, so they will live for long periods even after they've eaten all the larvae, waiting for more to hatch.
The kind of copepods proposed for mosquito control are referred to collectively as "cyclopoid copepods." Two kinds, Mesocyclops longisetus and Macrocyclops albidus, are the ones that behave most aggressively, destroying larvae that come near even when they don't eat them. 
About 25 years ago, they were first considered as a natural predator for mosquitoes. Tests have been done by adding them to water in containers, and so far, they've been fairly successful.
Recent trials showed they killed mostly first instar larvae of Aedes mosquitoes, but each copepod might destroy forty larvae per day, cutting Aedes breeding by 99 to 100 percent. They were less effective with Anopheles and least effective with Culex larvae.
Recent tests in Florida with the copepod Macrocyclops albidus in the lab and in field conditions showed similar results, with almost a 90 percent reduction in larvae.
In practice, they would best be used where most of the local mosquito problem is due to Aedes breeding in known locations, such as pools or water-storage containers. There are some slight risks with introducing copepods on a large scale, which scientists are still evaluating. Copepods themselves are associated with the spread of some diseases such as cholera, but they can't spread it unless the disease is already present in the area.
Live copepods like these are available for sale now as food for fish, but they're not ones that eat mosquitoes. However, the right species may soon be out of the testing phase and available for commercial sale as mosquito control.
Most insects are vulnerable to diseases caused by fungi, and several that target mosquitoes had been known of for years, but when scientists discovered Bti, other researchers became interested in looking for fungal diseases which might be a similar biological control.
The article "Entomopathogenic fungi for mosquito control: A review" summarizes research on various fungi which might be effective. According to the article, the difficulty is finding a fungus which is cheap to make and easy to apply, infects a high percentage of mosquitoes in field conditions, and stays active for several generations of larvae, while producing no harmful side effects to humans, the environment or other animals.
The three most promising kinds are Lagenidium, Coelomomyces and Culicinomyces, according to the authors. While each has some useful properties, none has all the attributes necessary to be a practical control.
Coelomomyces can kill many different kinds of mosquitoes, but is difficult to mass-produce. Culicinomyces requires high doses for control and doesn't last in the environment.
Lagenidium shows the most promise. Laginex, a product based on it, was even commercially marketed for a while in the 1990s, but never became a success.  Lagenidium targets mostly Culex mosquitoes, unfortunately not as high a priority for control compared to mosquitoes that spread malaria.
The authors of the article note that malaria mosquitoes will be more difficult to control with fungi, since they can breed in small, temporary puddles of water, such as ruts and footprints, so it would be difficult to spread fungi in all the areas necessary to kill their larvae.
Nematodes are microscopic animals which are usually harmful, causing illness in animals, but beneficial nematodes attack only creatures we want to get rid of, such as grubs or, in this case, mosquitoes.
Nematodes have the advantage of being safe for humans and other animals, as long as they target only a few harmful species. The kind of nematodes that have been most researched for controlling mosquitoes are called mermithids. Some of them target only one species of mosquito, or only one genus, so they're limited enough to be safe. The main problem with producing them as a commercial product is that they are difficult to keep alive during storage and transportation, which makes commercial sales impractical.
Two companies attempted to develop the nematode R. culicivorax as a commercial mosquito control in the 1980s. One result was a product with the apt name "Skeeter Doom," but neither company succeeded in bringing a nematode-based insecticide to market that was both profitable and practical. 
Future research may develop a way of producing and keeping alive nematodes that target mosquitoes specifically, but practical problems still need overcome.
Toxorhynchites mosquitoes, whose larvae are natural predators of other mosquito larvae, have been released occasionally in an attempt to control other mosquitoes. The idea didn't work when it was first tried in the mid-twentieth century, since the predators didn't reproduce well, and when they did, they often weren't at the right stage of development to kill the other larvae. Later attempts have shown more success, and there is still hope that Toxorhynchites may be useful as a predator.
You may wonder what good it does to substitute one kind of mosquito for another. Unlike most, Toxorhynchites adults don't bite!
Other small creatures which prey on mosquito larvae are flatworms and coelenterates, but storage and distribution are a problem in developing them for commercial use. Microscopic protozoa, also, are natural pathogens of some mosquitoes, but are difficult to produce commercially. 
Another method being tested is creating genetically altered mosquitoes. One possibility is releasing sterile male mosquitoes into the environment, in the hopes that they would mate with females, who would then bear no offspring. Another possibility is releasing males who have been genetically altered so they only sire male mosquitoes, creating a dwindling population of females. Mosquitoes may also be developed which are resistant to being infected by malaria or dengue. In that case, there would still be as many mosquitoes, but they wouldn't be able to transmit those diseases. 
As of now, all these forms of biological control each have problems preventing them from being brought to market, but they're all possibilities in the future.
- Growth hormone on the market now, prevents mosquitoes from maturing
- Mega-Catch Mosquito Trap reviewed--a CO2 trap that doesn't break down
- Do Mosquito Dunks really work?
 West Nile Virus Mosquitoes in Puget Sound Stormwater Systems Sergio Camacho.
 Larvicides and Larviciding, University of Florida Mosquito Information Website.
 Potential and Future, Nematodes as Biological Control Agents: Mermithidae by James J. Petersen.
 Biological Controls of Pests of Medical and Veterinary Importance
 Medical and Veterinary Entomology, by Gary Mullen, Richard Mullen and Lance Durden. Fighting Malaria with Spermless Mosquitoes, Harvard School of Public Health.
Copepod photo by Dr. Ralf Wagner, Wikimedia Commons. Toxorhynchites photo by Acrocynus, Wikimedia Commons. Culex mosquito image from the CDC.