But they have a problem. Although they are trying to blame the Democrats for the shutdown (because they refuse to repeal or defund Obamacare), the public knows why the government has been shut down, and which party is responsible for that. They know the shutdown was orchestrated by the House Republicans in a last-ditch effort to destroy Obamacare.
They know that the Senate voted on every continuing resolution the House sent to them (and rejected them), while the House has refused to allow a single vote on the continuing resolution sent back to them by the Senate. And they know why the House GOP has refused to allow a vote on a "clean" continuing resolution -- because they wanted a government shutdown to show their teabagger base that they are ideologically pure (and survive their own party primary next year).
And the American public does not like this tactic at all. A new Quinnipiac University Poll (taken between September 23rd and 29th of 1,497 registered voters nationwide, with a 2.5 point margin of error) shows an overwhelming majority of voters oppose the shutting down of the federal government to kill or hurt Obamacare. As the chart above shows, 72% of the general public opposes this shutdown, and that is true of every demographic of age, income, gender, and race. The only demographic without a majority opposing a shutdown to damage Obamacare are the Republican voters (and about 44% of them oppose the shutdown).
I don't know how long the Republicans will continue the government shutdown, but even if they were to abandon it immediately, they are already planning to do it all over again when the debt ceiling needs to be raised a couple of weeks from now. They have made it clear that they will refuse to raise the debt ceiling unless Obamacare is defunded, delayed, or repealed (among other things on their right wing wish list that they can't get through Congress).
These Republicans need to know that a significant majority of registered voters also oppose holding the debt ceiling hostage to harm Obamacare. This is shown in the chart below (and note, all demographic groups, except Republicans, oppose that also).