The US has decided to establish a US Marine presence in Darwin. What do benefits do both governments gain from this?
economic benefits inherent with hosting a military base
Even the opponents of the base (Australian Green Party) admit that:
The economic benefits of military bases include employment for local contractors and suppliers.
military benefits of both joint training exercises
better security, especially in light of much stronger Chinese geopolitical weight and power projection, coupled with Japanese resurgence as maritime power - the latter is a very long term concern, granted, but some people still remember the meaning of Greater East-Asian Co-Prosperity Zone.
USA gets a base in an important strategic location, in an extremely friendly territory, close to South-East Asia (next door to both Philippines and Indonesia, both countries with Islamist and Al-Quaida franchise presence) and closer to China. This is in line with Obama Administration announced 2011 strategic pivot to interests in Asia-Pacific region.
A pretty good list of strategic implications and benefits can be found in this comprehensive article by Gordon Arthur.
This is extremely important in light of issues inherent in stationing and basing in Islamic countries (if you recall, Osama Bin Laden's main original stated beef with the USA - outside of their support for House Saud - was the US forces based in Saudi Arabia)
To the answer of user4012 I would add an important point that is:
- Stricter control over the sea area shared by Australia, Timor East and Indonesia which contains important oil and gas fields.