The Shutdown seems increasingly likely, possibly multi-day. The Senate will bounce legislation back to the House with mere hours to spare; the House voting against it would ensure a shutdown. Republican leaders believe, or say they believe, that a shutdown will be seen by the public as the democrats fault, and hope that the removing Obamacare aspect will cement their place amongst their constituents. Privately, many Reps express doubt.
Polls are in showing the Shutdown is universally unwanted, from liberals to independents to conservatives to the Tea-Party. It is seen as both sides not willing to compromise, of the 2, Republicans are seen as the more inflexible and the target of most blame, with almost all of Democrats, 3/4s of Independents and over half the republican party blaming them primarily. Obama is seen as more flexible, but not overwhelmingly, with a significant portion blaming him over the republicans. Very few people hold both parties in equal blame. Obama's Approval has dropped slightly, to 43%. The Health Care law is still unpopular by a margin of 13.3% Approval/Disapproval, although one of the averaged polls had Fox news at a comparatively higher number (19%), so make of that what you will.
The upshot is that the voters will be unhappy with Washington as a whole, but Republicans stand to lose a lot more. This is only a early impression, if and when it happens the actual numbers may be fairly different.