Just today, a friend and I as Wilson and Woody in our day 160 world just managed to kill the Dragonfly for our first time in DST. I'm not here to brag, but rather to share the strategy we used in order for others to have the same success and joy that we did. Please don't patch this out...
This can be done with one person, but this is DST, so you'll likely have two people or more already. Character wise, you don't need any in particular although Wolfgang and Wicker are great for obvious reasons. The resources you'll require for the strategy aren't too expensive and if you're shooting for them could probably be gotten pretty early on in a world, as for the list, they're below.
6 Logsuits and Football Helmets (3 of each for the two of you, if you can get better armor, more power to you.)
6 Dark Swords (3 for each of you, but you could use 2 for each and a spear extra since after cracking 4 dark swords, she only has 200 left.)
Healing Supplies (My friend and I used 5 Healing Salves, 3 Trail Mix, and 4 Fishsticks for each of us, all pretty easy to get.)
A whole lot of Stone or better walls (80 will give you a whole lot of room to maneuver but you could probably do it with only 40.)
A Pan Flute (This one is real important. How you use it is details in the guide portion.)
Optional: A Walking Cane (Helps with the first step, you don't need it, but you really should have one anyways if you're going after Dragonfly.)
Optional: 2 Miner Hats (So you can continue the first part of the guide during the night.)
Optional: Campfire/Endothermic (In case of night and also temperatures during Winter/Summer.)
Optional: Snacks (Don't Starve!)
As for the battle, I've separated it into 3 phases to do in order for success.
Phase 1: As soon as you and your friend(s) get to Dragonfly's arena, have one person who preferably has a walking cane get dragonfly's aggro and lead her around in circles away from everyone else. While this is happening, everyone else should begin to construct a box in the middle of the arena made of stone walls. Leave one gap in the wall so whoever is being chased can come into the box as well.
Phase 2: Once the box has only one gap in, lead Dragonfly in and get your armor and dark swords out and begin swinging away. Watch your HP, and if it gets too low for comfort, eat either your trail mix / fish sticks / other good HP food. She should eventually get stunned and continue swinging for as long as you can, as soon as she flies off towards the pools, use your healing salves / honey poultice to get yourself healthy again as you can't use them while Dragonfly's attacking you, then go to phase 3.
Phase 3: Put a wall into that little gap so you're completely walled in. Once Dragonfly has spawned all her larvae, don't get too close to the walls or they'll hit you through them. (hence why you want a big box for the strategy.) Fight her like you did in phase 2, but keep an eye on the larvae. As soon as they all die, she'll become enraged and probably do her stomp attack, ruining the entire plan. There's a delay between the last larvae dying and her becoming enraged. During this time, use the pan flute to put her to sleep. This will stop her from getting enraged and give you an opportunity to heal up and get ready to continue attacking. Rinse and repeat until she's dead, and celebrate over your newfound rewards.
Anyhow, this may not be the best or most efficient Dragonfly strategy, but it's the simplest method my friend and I could think of that also required the least amount of resources total. As for the pan flute's durability, Dragonfly usually drops a green gem to help renew it. Best of luck out there against Dragonfly.