Requirements: Be Injured At the Very Beginning.
To start off, you can buy a lantern for 20 coins from almost any town. Then head to the Swamp Village, the town west of the swamp and northeast of the starting location. Go into the city and talk to the Doctor inside his office. Let him heal you, this must cost you at least several coins. Talk to him again, this time ask about his son. He will say that is son is deathly ill and needs medicine; he wants Nite Berries to subside the pain. But the Doctor cannot simply leave his office; he asks for your help - accept the quest.
Leave the city and go south until you spot a ranger named Damon. He says that you need Swamp Boots to traverse the dangerous and murky swamps. He will give them to you. Now head east into the swamp. You will encounter monsters such as Level 26 Alligators and Level 3 Frogs. Following the patch you should see a dock.
If you brought a boat, instead of the 20 coins, go onto the dock and head east until you see another and land it; you will be on the island.
If you brought 20 coins, head south of the dock until you see a Swamp Shack. Talk to the guy inside, who will offer you a trip of 20 coins to pay for the trip's fuel. Give him 20 coins and you will arrive at a island at a dock.
On the island, grab one Nite Berry. Then go down the pit with your Lantern, inside you will find more alligators. Once inside, go south and then west. You should now be at the very bottom of the dungeon. Click on the outer edge of the dungeon and you should see "Examine Wall". It asks if you want to hit it with your pick axe; select yes.
When you hit it, you will uncover a new path leading to swamp water. Click on the swamp water to fill it up with your bucket. Quickly press Reset or return to the top and talk to the person in the Swamp Shack on that island and give him 20 coins/use your boat to return to the other island after you make sure you've filled it in.
Head back to the Doctor and give him the Nite Berry. Talk to him again. He will be very happy; but the boy is still somewhat ill. He asks for swamp water that might finalize his cure. Give him the swamp water and the boy will be cured. The Doctor is pleased with your work and now the quest is complete.