[Papaya]

To extrapolate on that idea...

Spoiler for My own theory:

Judging from what Ryuk said, Shinigami "no longer" have any purpose in the world. This seems to be a strong sociological statement to me. When that statement is taken in conjunction with what Bahamut89 said, you can make conjectures about why Shinigami kill. On particular reason that could provide impetus for killings is that they are keepers of order. Evidence is as follows:
1. They must kill to stay alive.
2. They cannot tell what a person will do (they are not omniscient).
3. They are selective in their targets, though they most likely do not kill children, who would give them the longest lifespan additions--if they did that, they wouldn't need the Shinigami's eyes.
4. Assuming that, we can say they are not evil beings, and may have originally killed evil beings.
5. The portals from the Shinigami world see numerous facets of Earth; this could be to see when people were "sinning".
6. The Shinigami's eyes are partly a tool to see how fast you could stop a sinner's life before he does anything worse.
In that theory, Shinigami were first-action enforcers. I'm guessing that Ryuk's allusion to "no one cares anymore" to the author's implication that the world has become largely immoral and its ethics completely convoluted, which fits the overall theme of the series.