Australia is home to an American military base and for that reason I do believe we should maintain some troops within Australia. American military bases can be found around the world and I find that Australia is actually an important place for us to keep troops given it's location in the world.
Every nation should have a prescence in the other countries of the world. Embassies are common, so why not troops? In the case of emergency, American troops are valuable and can help the nations they are stationed in. Natural disasters would be helped by these troop, and the diplomatic benefits are great.
While it may make logistical sense for the United States to keep bases on foreign soil to allow for refueling and resupplying, there is no need to keep troops in Australia. The United States, the United Kingdom, Canada, and Australia are all so closely aligned in their geopolitical goals that it is nearly impossible to imagine a situation in which the United States would be denied use of Australian facilities (manned by Australians) in a crisis. American military resources would be better spent elsewhere.
I don't think there is any pressing reason why the United States should have troops in Australia, Australia is not under threat by any large nation and Australia itself should be very capable of defending itself without the assistance of troops from the United States, so in general I don't see a valid reason for there to be troops in Australia.
Australia is an ally, and we should not be sending troops over there for vacation stay if they are not doing anything. I have nothing against any of our troops, but there is not any unrest in Australia as there is in the entire Middle East and parts of Africa.