Some leading Republicans have rebuked Trump, distancing themselves and the party from his views.
"This is not conservatism. What was proposed yesterday is not what this party stands for," House Speaker Paul Ryan (R-Wis.) said a day after the presidential candidate proposed barring Muslims.
At the same time, other Republicans have been circumspect in their criticism of the party's presidential front-runner. Critics claim these GOP politicians are blowing a "dog whistle," inaudibly appealing to their constituents' prejudices.
The same argument has been applied to almost every recent presidential election: President Ronald Reagan was criticized for invoking the image of the welfare queen to imply that African Americans were lazy crooks who bilked the government. President George H.W. Bush's infamous Willie Horton spot told the story of a black man who committed violent crimes while on furlough from prison, a move that critics say won Bush supporters by exploiting white Americans' fear of black crime.
An expanding body of research by psychologists, economists and political scientists suggests that voters' racial biases help the GOP win elections, and critics say the party is capitalizing on that fact. Though researchers haven't settled how successful dog-whistle politics are at tapping into those prejudices, some believe that race will become more, not less important in the party's future campaigns.
"There's a good deal of evidence that white resentment of minorities is linked to support for Republican candidates, their policies and conservative ideology in America," said Robb Willer, a political psychologist at Stanford University.
On the campaign trail, reporters frequently bring up Trump's rhetoric and ask the other candidates to make their positions clear at a time when white Americans are a rapidly declining portion of the population. Bruce Bartlett, who served as a senior economic official under Reagan and George H.W. Bush but now describes himself as independent, said Trump is giving Republicans a crucial opportunity to win over a larger, more diverse electorate by repudiating prejudice.
"Trump is forcing Republicans, at long long last, to finally decide, 'Are we going to be the party of racism and lose the White House forever?' " Bartlett said.
As the country has become more diverse, the Democratic Party has, too. But the demographics of the Republican Party have not changed much in recent years, according to Gallup. As of 2012, 89 percent Republicans were non-Hispanic whites, compared to 60 percent of Democrats. Nearly three quarters of Hispanic and Asian voters and fully 93 percent of black voters cast ballots in favor of President Obama in 2012, according to Washington Post exit polls.
Research has shown that voters who favor Republicans are more likely to hold racial biases against people of color. For instance, nearly one in five Republicans opposes interracial dating, compared to just one in 20 Democrats, according to the Pew Research Center.
A poll conducted by the Associated Press before the 2012 election found that 79 percent of Republicans agreed with negative statements about racial minorities, such as "If blacks would only try harder, they could be just as well off as whites." Among Democrats, the figure was just 32 percent.
The Republican National Committee did not respond on the record to inquiries about Reid's statement.
'The white man's party'
The data is all the more surprising at first blush since Democrats were the party of segregation in Southern states for decades. That began to change in 1963.
On June 11 of that year, President John Kennedy, a Democrat, gave a televised speech in favor of racial equality. That spring, the share of white Southerners who approved of Kennedy declined by a precipitous 35 percentage points.
Some Republicans saw an opportunity. By emphasizing their support for "states' rights" and "law and order," they could subtly appeal to those disaffected white voters, showing their support for the status quo without explicitly opposing civil rights.
"Substantial numbers of Party leaders from both North and South see rich political dividends flowing from the Negrophobia of many white Americans," wrote journalists Rowland Evans and Robert Novak, both well sourced in conservative circles, two weeks after Kennedy's speech. "These Republicans want to unmistakably establish the Party of Lincoln as the white man's party."
Over the next few decades, white support for Democrats in the South — once monolithic — gradually eroded.
Bartlett, the former Reagan administration official, argued that white Southerners did not leave the party because of racial bias. He said these voters held conservative positions on issues unrelated to race, such as health care and Soviet relations. That group gradually realized the Republican platform matched their views more closely.
[Read more: How racism explains Republicans' rise in the South]
On the other hand, a new paper by economists Ilyana Kuziemko and Ebonya Washington rebuts
alternative explanations for the decline in the number of Southern white Democrats. Analyzing archival polls, the authors found no evidence that racially conservative white Democrats who left the party were more conservative on issues unrelated to race than those who stayed in the party. The only explanation for their desertion of the Democratic Party that was consistent with the data was racial animosity and opposition to civil rights.
While some leading Republicans have tried to put this history behind them, there's reason to think the problem of prejudice will return with a vengeance in this campaign and in subsequent elections.
Maureen Craig and Jennifer Anne Richeson, psychologists at Northwestern University, recently conducted an experiment in which participants read about the fact that white residents are no longer the majority of California's population. Reading this information made white participants more likely to support the Republican Party and more likely to endorse conservative views in a questionnaire.
Given that white Americans' share of the population is declining in many other states, too, white voters could be swayed by candidates who talk about race, openly or not.
"Race is inevitably going to become a more and more important part of Republican strategy as it becomes more and more a party of whites and especially white men," said Eric Knowles, a political psychologist at New York University. "Demographically, that's where their base is."
Knowles's research suggests that this trend has already begun to affect GOP politics. Using data from a series of surveys, he and his colleagues found that membership in the tea party increased respondents' perceptions of white identity over time. Other researchers have found that its members tend to be more racist and xenophobic, after accounting for their belief in limited government and other conservative principles.
In other words, there are many Americans with conservative views, but those who also hold prejudices against foreigners and people of color were more likely to join the tea party.
"The shift we're seeing right now in politics is really an unselfconscious and unabashed reintroduction of racial interest and racial rhetoric in campaigns," Knowles said.
Those tea party studies were conducted nearly five years ago, and data from the current campaign isn't yet available. To some observers, though, the Republican field's rhetoric fits a familiar pattern.
Trump is "the walking id of the Republican base, and maybe of the American people, too," Knowles said.
The real-estate magnate's proposal to bar all Muslims from entering the United States is a grandiose example, but his contenders are employing similar tactics.
In response to Trump's plan to ban Muslims from entering the country, Sen. Rand Paul (R-Ky.) said this week, "I've called for something similar." Paul said he opposed a test for immigrants based explicitly on religion, but added that screening immigrants based on the risk they posed to national security would accomplish the same goal.
Neurosurgeon Ben Carson has also suggested that Muslims might be dangerous, comparing the predominately Muslim Syrian refugees to dogs.
"If there's a rabid dog running around in your neighborhood, you're probably not going to assume something good about that dog," Carson said last month. "It doesn't mean you hate all dogs, but you're putting your intellect into motion." He has also said he does not think that a Muslim should be allowed to become president of the United States.
As for Trump, the one issue that appears to unite his supporters is opposition to immigrants. A majority of Republicans who support the deportation of undocumented immigrants and oppose accepting refugees from Syria also support Trump.
Some politicians and pundits have called Trump a fascist and a demagogue, but the term wrongly suggests that his approach is radically different from that of other Republican candidates, said Ian Haney-López, a legal scholar at the University of California, Berkeley.
"These sorts of terms make Trump seem as if he's this exceptional unique outlier, that he's doing something that nobody else has done," said Haney-López.
"Clearly, in some ways he's different from other politicians," Haney-López said, "but in his strategic decision to pursue support, to mobilize support by appealing to people's racial fears, he's well within the tradition that has been established in the Republican Party since roughly 1963."