Mosquitoes can mostly be ignored. They are rather difficult to hit successfully, and are incapable of dealing more than the slightest damage.
Mosquitoes will not attack players possessing the Insect Incense.
Mosquitoes offer almost nothing of value to the player. They may drop items, but their ability to evade attacks makes mosquitoes a poor choice for acquiring items.
A variant of the mosquito, the Metal Mosquito, appears in the Greenhouse. It has higher attack power than the normal mosquito, but is otherwise statistically identical and is not much more of a threat.