Yeah, that's the difficulty. In fits of scratching and waving my arms on camping trips of yore I too have felt the urge to just pour a trillion liters of pesticide over the world and kill all of the little flying bastards in one fell swoop (har har). Problem is, it's actually really difficult to imagine such a plentiful and prodigious species _not_ playing a major food chain role. They're everywhere, and things that eat bugs are everywhere; it would be really hard to imagine their absence not collapsing some part of the food chain.

The only thing that would make sense to me would be to introduce some "invasive species" that eats the mosquitoes, doesn't spread the malaria, and can still be eaten by most of the things that eat the mosquitoes. Still, introducing any invasive species is playing with fire; it's literally impossible to know all the consequences in advance.