The United States doesn't need troops in Australia, but it would be nice to have a foundation next to China and Japan. This may have been an obvious yes during WW2, when it took a lot longer to transport troops and goods overseas. Having a small number of troops in Australia might send a message to China, and help us if we needed to set up a home base during a war.
There's nothing in Australia itself that poses an immediate threat that I'm aware of, nor as has been previously mentioned is there a need to have troops positioned around the world for quick response anymore with the technology we now possess. I can't think of any specific reason we need troops stations in Australia at this time.
The US keeps troops abroad with the idea that if they need to mobilize quickly they will have troops that can get their in no time in any part of the world. However, this is no longer necessary as the US's tactics have developed into using drone and technological warfare, rather than physical soldiers. The fact that troops are still their is just a throwback to wars that have been long gone for decades, and they are now only wasting money.
After a pair of wars in Iraq and Afghanistan, the last thing that the United States needs to do is to keep troops abroad for no reason. Australia has shown themselves to be a country of the utmost stability. If they need help down the road, we can show up.