Despite the government shutdown that has idled most of NASA, the Mars Curiosity rover will still remain in operation, according to an October 1, 2013 story in The Verge. This is because the Jet Propulsion Laboratory, from which the Curiosity is controlled, is a private contractor run by the California Institute of Technology and is thus not subject to the government shutdown, at least for now. However the JPL website is not being updating and the JPL twitter feed has been taken down.
Large parts of NASA have been shuttered, however, thanks to the shutdown. One of the only departments left in operation is Mission Control at the Johnson Spaceflight Center south of Houston which monitors activities on the International Space Station. Most other NASA projects have been suspended, particularly the preparations for the upcoming launch of the MAVEN Mars orbiter. If the MAVEN is delayed past its late 2013 window it will have to go back into storage and wait until early 2016 before it can be launched. MAVEN is designed to study the Martian atmosphere.