According to Louie Helm eradicating a species of mosquitoes could be done for as little as a few million dollar.
I don't have a few million dollar lying around so I can't spend my own money to do it. On the other hand, I think that on average every German citizen would be quite willing to pay 1€ per year to rid Germany of mosquitoes that bite humans.
That means it's a problem of public action. The German government should spend 80 million Euro to rid Germany of Mosquitos. That's an order of magnitude higher than the numbers quoted by Louie Helm).
The same goes basically for every country or state with mosquitos.
How could we get a government to do this without spending too much money ourselves? The straight forward way is writing a petition. We could host a website and simultaneously post a petition to every relevant parliament on earth.
How do we get attention for the petition? Facebook. People don't like Mosquitos and should be willing to file an internet petition to get rid of them. I would believe this to spread virally. The idea seems interesting enough to get journalists to write articles about it.
Bonus points:
After we have eradicated human biting mosquitoes from our homelands it's quite straightforward to export the technology to Africa.
Does anyone see any issues with that plan?
No, I am not.
I personally rate the probability of catastrophic negative effects of this action as significantly lower than the probability of any negative effects, and the probability of no negative effects.
I am also speaking from a position of ignorance, and I don't like making decisions from ignorance.
I don't know what the ill effects would be, but the benefit is clear. If more cheap information is readily available, I want it (and some has been provided). If some amount of expensive evidence would increase my estimate of catastrophic effects, and that evidence can be clearly defined and gathered, I want it. If only vague, hard to measure risks remain, I say do it.