"Poll Shows Support for Embattled Public Sector Workers" is the headline over the New York Times' news article about its own poll. If you actually read the poll, it says that 22% of those surveyed in the poll are either public sector workers or their household members. Back them out of the poll, and one could hang a different headline over it.
That's not the only way the poll is skewed. One question reads in part, "Some states are trying to take away some of the collective bargaining rights of public employee unions. Do you favor or oppose taking away some of the collective bargaining rights of these unions." This question measures how people feel about "taking away" "rights" — words repeated for emphasis — as much as or more than how they feel about collective bargaining for public employees. You'd expect people to be reflexively opposed to taking away rights.
Another question asked, "If you had to choose one, which of the following would you be willing to do in order to reduce your state's budget deficit 1. increase taxes 2. decrease benefits of public employees 3. decrease funding for roads 4. decrease funding for education." The spending cuts all have specifics attached to them, while the tax increase is unspecific and abstract. If the pollster really wanted to weigh taxes and spending, the poll could have asked, "increase the income and sales tax you pay." If the spending cuts had been phrased as "save money by having public employees contribute to their health care and pensions in the same way that private sector workers do," rather than "decrease benefits of public employees," that answer would have been more popular. The way it was phrased, respondents might think the benefit of a public employee was that your children learn to read and criminals get arrested, rather than something to do with pensions or health insurance. "Roads" is a straw man — most states spend more on health care than on roads, and if "find efficiencies to spend less on Medicaid health care for the poor" were a choice, that might have been more popular than raising taxes, particularly if it was people's own taxes being raised. The question even avoid the word "spending" altogether — it's unpopular — going with the more anodyne euphemism "decrease funding."
Anyone who concludes from this poll that the Republicans are making some kind of huge political error by targeting spending rather than raising taxes would be making an error.