Generally speaking, it seems like a bad thing when human beings eliminate an entire species. That said, it is usually done by accident, as a consequence of habitat destruction and pollution. To a considerable extent, we should probably scale back those harmful activities, and think about backing up some DNA in the meantime.
In at least one case – the eradication of smallpox – the destruction of a species seems unambiguously excellent. Indeed, it is a shame we didn’t manage to finish the job, and that Russia made such huge quantities of smallpox as a weapon. Mosquitoes are another candidate for a species we could wipe out without guilt, especially since ecologists are arguing that they don’t serve a major ecological role.
The ethical question is: if it were practical to do so, should humanity exterminate all mosquitoes?