"I think that the law has never really recovered from the bungled rollout of the health exchanges," said Whiteman, referring to the botched performance of the federally run HealthCare.gov and several of state Obamacare exchanges when they launched last fall. Since then, HealthCare.gov has been repaired and performed well in the last five months of operation, as did a slight majority of the 14 state-run exchanges.
Whiteman also said the continued "negative political messages" by ACA opponents, particularly House Republicans who have repeatedly tried without success to gut Obamacare, "has to be having an impact on people's opinions."
Read MoreObamacare hasn't cured drugstore sticker shock
Bankrate's study found that people who earned more than $50,000 annually were most inclined to have negative feelings about Obamacare's effect on the country. However, people making $30,000 or less were inclined to be positive about the law.
When asked about Obamacare's effect on their own lives compared to last year, 52 percent said they felt it was about he same. But among the rest of the respondents, people who felt more negative about how Obamacare directly affected them since last year outnumbered those who felt positively "by about 2 to 1," Whiteman said.
Bankrate also questioned respondents for the first time about a still-small, but growing trend of companies moving workers out of employer-provided health insurance and into individual plans sold on private exchanges, which are similar to the government-run Obamacare markets.
Earlier this month, Standard & Poor's Capital IQ estimated that 90 percent of S&P 500 companies could move their workers toward such private exchanges by 2020, saving an estimated $689 billion in costs in the process.
And the same report found the potential for a stunning $3.25 trillion in savings if all companies with 50 or more employees made such a move.
Most large companies that shifted workers to private exchanges continue subsidizing, in some way, the cost of health coverage, while offering workers increased options in number of plans and level of coverage.
Bankrate said 30 percent of respondents believed such a shift toward private exchanges would have a negative impact. Just 14 percent said it would be a positive move.
—By CNBC's Dan Mangan