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Title says it all really? Will the staff operating Curiousity, Insight, Juno, etc. continue to work? What about staff supporting them indirectly?

There's a lot of good stuff at the link provided by Katie, particularly the link to the NASA revised shutdown plan memo.

Personnel deemed essential need to report to work as usual. They don't get paid during the shutdown but are guaranteed backpay once funding is restored. Any active flight mission would have its personnel identified as essential. Some ground tests might identified as essential, though that is more difficult to justify.

Many science missions are operated by contractors. JPL is an FFRDC managed by Caltech, so employees are contractors for NASA. They are not affected by the shutdown as long as Caltech/JPL has money in the budget. Curiosity, Juno, and Insight are all operated by JPL, so there should be little impact on direct support personnel as they are not affected by the shutdown. They come to work and get paid normally.

Any NASA employee directly supported ISS operation would be essential.

See Casey Dreier's post in the Planetary Society blog: Happy Holidays. NASA is Shut Down.

Here are some of the key points:

NASA is among the federal agencies covered by seven appropriations bills that have yet to be passed by Congress and signed into law for fiscal year 2019. The remaining five bills covering the departments of Defense, Energy, Heath and Human Services, and Veterans Affairs (among others), were approved earlier this year. This is a partial government shutdown that will still impact approximately 800,000 federal employees and disrupt many services. It is essentially a function of bad luck that NASA's appropriations were held up, grouping it with the remaining agencies and the Department of Homeland Security, the epicenter of this particular political battle.

Since this is the third shutdown this year(!) we can regurgitate some of our past coverage of what happens to NASA projects and programs during this absurd situation. NASA has also released its guidance for programs impacted by the shutdown (PDF).

Operations of the ISS will largely continue unaffected, as will the New Horizons flyby of 2014 MU69 ("Ultima Thule")—though NASA's public relations team will be absent. John Hopkins' Applied Physics Laboratory, which manages New Horizons for NASA, will still cover and publicize the flyby.

NASA will continue to communicate with and safely operate its uncrewed spacecraft. Science planning and operations may be impacted without the participation of federal scientists, and it is unclear how long regular operations could continue during an extended shutdown.

Missions in development (such as Mars 2020 or the James Webb Space Telescope) can continue if they are being led by contractors. Those contractors (including JPL) have some cash reserves and will continue to until the cash runs out or they reach an approval checkpoint that requires NASA sign-off. In a previous lengthy shutdown, Caltech, which operates the Jet Propulsion Laboratory for NASA, temporarily put up its own cash to keep the doors open. Again, the degree of impact here depends on the duration of the shutdown.