The danger for Republicans is that the budget cuts will severely weaken public support for the austerity theme that the party has been promoting since 2010. The cuts will make "deficit reduction" something very real to average American citizens and business and something that is often quite painful rather than an abstract debate over numbers.
While Americans have historically been hostile to government, they tend to support specific government services when asked by pollsters. So Washington's overall spending might not be popular as a concept, but Social Security and Medicare are.
The spending cuts will shift the debate toward the specifics. Americans will watch as government services are retrenched. The last time this happened, things didn't go well for the GOP.
That is, barring some grand and hidden scheme by Republicans that will successfully paint Obama as out of touch with the average American. That will be really hard, as it is Republicans who have driven the sequestration dialogue. Obama has helped egg them on, to be sure, but has taken great pains in his statements to set up a "you break it, you own it" theme.
After all, it's the Teabaggers who have run and won office on the promise to cut spending and keep taxes low, and they got their lunch stolen when Speaker John Boehner's caucus caved at the end of last year. Despite the high profile Weaker Boener has given to the Teabaggers (the State of the Union address featured not one but TWO responses from Teabaggers) there's still a lot of anger and the urge to drive off the cliff is strong.
Thus, we're seeing a lot posturing this week, starting with last night's fiendishly timed (just before the Oscars® aired) release of the list of how budget cuts would impact each state, starting with furloughs for Federal employees and drilling all the way down to the number of children who will not receive immunizations in Georgia (as an example.)
Many if not most of these cuts wouldn't be felt for some time. For instance, Federal law requires 30 days notice to lay off Federal employees, so staff cuts won't happen until April 1 at the earliest. Some impacts will be felt immediately, like Social Security checks not being mailed out for the 3rd of March, of course.
To be honest, this all seems like some Grand Guignol opera: the Teabaggers get their fiscal cliff and government shut down but then the responsible adults step in almost immediately to restore balance to things.