Will the January 2018 government shut down affect the NSF postdoc awards?

The NSF recently changed its accounting practices, so that money for awards are committed all at once, rather than year-by-year. So, that means that the funding for your award should already have been “available” when you got the notification. Unfortunately, a shutdown means that no further processing will likely take place until it ends, so there may be some delays in the start of your award. But I can’t imagine a scenario where the NSF cancels awards it was planning to make because of the shutdown.

Congrats on the NSF postdoc! As Brian Borchers' comment indicates, none of us can answer this question definitively, since we aren't the NSF (or the U.S. Congress for that matter). Arrangements for the 2013 U.S. government shutdown (which as you may recall was in October and lasted over two weeks) resulted in submission deadlines being postponed, and in NSF Postdocs (at least in math, and I presume in other fields) not being paid for the duration, which contrasted with NSF graduate fellowships, which are paid to schools in lump sums in advance. (As an aside, this has tax implications you should be aware of when you accept the award.) Awards (and selection for awards) resumed after the shutdown and back pay awarded for affected individuals. (Excuse the lack of references throughout, normally I'd be linking to the NSF website, which of course is currently displaying only minimal information.)

All the paperwork required for an NSF postdoc to be official takes months (requiring proof of graduation among other things) and it sounds like this shutdown caught you at the very beginning of it; the next steps will probably be delayed with the shutdown, but speculating (as nothing more than a member of the public) unless there is a major change to the NSF budget or a very long shutdown, it's probably not time to panic yet. ;) It is worth understanding that future shutdowns, should they occur after you start your postdoc, could affect your paycheck and when it arrives.