In this page of the guide to Animal Crossing New Horizons, you will learn whether the wasps are dangerous, should you avoid them... and is it even worth the trouble?
Let's start from the beginning - you can encounter wasps only in one situation. They will appear if you drop a hive while shaking a tree or using an axe on it. Angry insects will try to catch and bite you.
Nothing will happen if you let them do it. There is no health bar in the game. Except for some bite marks there won't be any consequences of the incident. Every encountered citizen of the island will be worried about your state, but it won't impact the gameplay.
It is possible that the wasps will bite you the second time in a row. This will also not affect the gameplay - the only difference is this time you will wake up in front of your house.
A bitten face, however, can disturb us. A medicine can help with that. Once we apply it, our face will return to normal state. Where do I get the cure? Medicine can be bought (from Nook brothers) or crafted. To create it, you will need a hive (obviously) and three weeds.
PROTIP - if you have a butterfly net equipped, then a quick reaction may save you from being bitten when a wasp nest falls to the ground. You can catch wasps. This way you will not only avoid bites, but also be able to sell or give the caught specimen to the museum.