Fortunately, we have lots of practice in controlling them
Researchers agree that global warming will increase the number of mosquitoes, but they disagree whether we can do anything about them and, more importantly, about an outbreak of mosquito-borne diseases.
Global warming is controversial in general, not just on the topic of mosquitoes. Even readers of children's books divide radically into both one-star and five-star reviews for the same book.
Adult mosquitoes like temperatures above 50 degrees, so there's no doubt more mosquitoes will be active if the average temperature rises. They also need wet conditions to breed, and global warming can create more of that in some areas, even if it makes other areas drier.
Some of the scarier scenarios were summaried by Sheila Dichoso of Northwestern University in a post reporting on the 2009 conference of the American Association for the Advancement of Science. There, experts on the environment and public health warned that diseases have increased since 1970, including West Nile virus, malaria, yellow fever and the plague, all spread by mosquitoes.
But not all experts agree. More recent research indicates that it takes more than warmer, wetter conditions to spread disease, even if there are more mosquitoes.
Although malaria and similar diseases are more common in the tropics today, entomologist Paul Reiter argues that what really matters, in stopping malaria, is human intervention.  People have learned to screen out mosquitoes, get rid of standing water, drain marshes and swamps, and spray mosquito habitats, as well as give better medical care to those suffering from the disease.
that will be helped by global warming.
Peter Gething published information about the historical incidence of malaria in the journal Nature in 2010, showing that between 1900 and 2007, there had been a decrease in the disease, in both its range and intensity, even though global temperatures had increased.
The southeastern U.S., which has always been warm and humid, had period outbreaks of malaria in 1900, but that no longer happens today. The hardest place to fight the disease is central and western Africa, where there's widespread infection of malaria in the population and mosquitoes bite year round. But Gething and his co-authors argued that climate change is not the main factor in the spread or control of malaria.
Encephalitis carried by mosquitoes also increases when temperatures and rainfall increase, according to Michael Gale Moore's book, Climate of Fear. But diseases like yellow fever and dengue, as well as malaria, used to occur regularly in the United States, and now they're rare, despite warmer temperatures, so we have learned to control them.
He cites an example of an outbreak of dengue in 1995 in the Caribbean, Mexico and Central America, with 74,000 cases of the disease reported. Some areas, however, were barely affected, while people living nearby suffered from the epidemic. "The contrast between the twin cities of Reynosa, Mexico, which suffered 2,361 cases, and Hidalgo, Texas, just across the border, is striking," he writes. "Including the border towns, Texas reported only 8 nonimported cases for the whole state." Better sanitation, education, elimination of standing water, window screens, and other control methods made the difference. 
The U.S. Environmental Protection Agency, citing information from the World Health Organization, is also cautiously optimistic, noting that climate change can provide new opportunities for mosquito-borne diseases, but the actual increase "will depend not only on climatic but also on non-climatic factors, primarily the effectiveness of the public health system." 
Mosquitoes are Adapting
But even if global warming doesn't cause an upswing in diseases, it will probably cause an increase in mosquitoes which will need controlled. Mosquitoes have survived millions of years because they're adaptable. Scientists have already noticed how they're responding to climate change, to increase their survival.
Pitcher plant mosquitoes, Wyeomyia smithii, spend their winters hibernating as larvae in water trapped inside the pitcher plant. They use day length, rather than temperature, to signal when to go into hibernation.
As the climate changes and winters get shorter, the larvae who take advantage of a few extra days of warm weather before hibernating are more "fit" for the new environment. Scientists have noticed those larvae do survive better, and therefore the hibernation period of pitcher plant mosquitoes has changed because temperatures have become warmer. "Between 1972 and 1996 mosquito populations at 50 N latitude evolved to wait 9 days later to go dormant," according to an article on microevolution at the Understanding Evolution website. 
So there is no doubt that mosquitoes will be able to adapt and thrive, as global warming creates temperatures which favor them. The good news is that humans have had decades of success in learning how to control them, as well as the diseases they spread. If we apply what we've learned, we should be able to prevent disaster in the long run.
 "Global warming and malaria: knowing the horse before hitching the cart," Paul Reiter, 2008.
 "The Health Effects of Global Warming," Climate of Fear by Thomas Gale Moore.
 Climate Change--Health and Environmental Effects, U.S. Environmental Protection Agency.
 Examples of Microevolution
Public domain photos courtesy of the CDC.