Eh, I really don't care if I get stung. If I manage to catch the bee, I can get some good pocket change from both the bee and the hive. If I get stung, I can go to Phoebe for medicine or just save+quit and my bee sting will be gone. No big deal. :3
I don't any more because I chopped down all my trees for landscaping x) But before, I didn't care. If I got stung, I had an Uchi neighbor to give me medicine. The Bells and furniture were totally worth it B)
I used to really be scared of bees, especially in the older games. I temporarily came over it in order to catch a spider in City Folk (I managed to catch a bee, too). I'm not scared so much anymore because if a hive drops, I can just save & continue and they'll be gone. (I'd probably be too scared in another version, though...)
I'm in the process of landscaping, so I don't have enough trees that drop bees/furniture/money, although I used to shake them and catch the bees for selling. I honestly don't mind the bees at all in NL since they're infinitely more catchable than in, say, WW. It took me nearly a year to catch my first bee in that game.
If I need Bells and actually go to shaking trees, well...I just love the chase. XD I love laughing at the bees by disappearing into the nearest villager house or Public Works Project and leaving them stunned. c:
I think it is only worth it if you are really bored and have nothing better to do! You can open your gates to prevent some features like bee hives dropping