When Luffy was young, Shanks saved him from the Lord of the Coast, who nearly ate Luffy. This cost Shanks his arm. Then we see the Lord of the Coast still inhabits the area 10 years later.
Why was this dangerous sea king not dealt with by Shanks or Garp during this time?
This is a bit of a strange question, but basically for the same reason any shark or bear is allowed to live in the real world. The Lord of the Coast didn't just bite Shanks' arm and scare Luffy, he killed and ate Higuma the bandit, and probably others as well. But because he lives in the water, he's not easy to find and kill, and neither Garp nor Shanks are the kind to want to put Luffy in a completely protected environment.
Even if someone did find and kill the Lord of the Coast, they've hardly made the ocean safe for children, there are plenty of other Sea Kings and sea creatures out there, and it's even possible the Lord of the Coast was actually making Foosha Village a little safer by occasionally eating some unpleasant pirates headed to the island.
It's also worth noting, Luffy almost certainly didn't kill the Lord of the Coast, just punched him out of his way and moved on. He is probably still in the area, traumatizing or killing the crews of other small boats.
