Without knowing, I assume that the island in question is Kangaroo Island, a popular tourist destination for South Australians (you can take a ferry if you wish).
The reason for the overpopulation is that koalas are not native to that island (it is pretty big—I’m guessing about the size of Puerto Rico), but were relocated there from the mainland when koalas were thought to be endangered.
As expected the koalas thrived—too much so, as it turns out.
When I lived in Glenelg, you could take a ferry directly there for a short time. This ferry was not able to compete with the one that left from Cape Jervis (and I think still does).
An hour and a half drive (or so) from Adelaide and maybe an hour on the ferry—you can drive right off and relax. Personally, I would not spend the time looking for koalas, but head for the south shore of the island and the sea lions. They are a lot easier to spot than those little guys hiding in the tress, looking pretty much like a part of the tree.
I was just watching a documentary on a particular bird in either Australia or New Zealand (I forget exactly), but they said that there was a bird that was so numerous that they were killing it freely. It finally got the point where they made it extinct.
Hopefully these professionals are the only ones doing the shooting and not some rednack with a truck, going around shooting wildly like Uncle Jimbo - "THIN OUT THEIR NUMBERS!!!!!"
Don’t forget guys, that it was the introduction of koalas to Kangaroo Island that created the imbalance. In this case if ‘man’ were to eradicate all koalas from the island, the balance would be restored.
Note that I am not necessarily advocating the eradication of koalas from the island—only making the observation that the initial intervention was the introduction of a non-native species. We just usually think of these mistakes as being things like cane toads (and I can assure you that most Aussies would be foursquare behind their eradication from Queensland.