It is not just Mr. Trump’s anti-immigrant, anti-Mexican and anti-Muslim remarks that have led others in the party to point to him as damager-in-chief of the party reputation. It is that his comments are exactly the opposite of the party’s self-stated goals for this election. After Mr. Romney’s 2012 defeat, the party convened a task force to examine its recent presidential losses. The result was a report on the party’s growth and opportunities among the electorate. In the Growth and Opportunity Project, the task force wrote: “It is imperative that the RNC changes how it engages with Hispanic communities to welcome in new members of our party. If Hispanic Americans hear that the G.O.P. doesn’t want them in the United States, they won’t pay attention to our next sentence.”

The prediction could not have been more accurate. Most Latinos didn’t think the party thought very much about them one way or another in 2012. But after months of hearing the party front-runner talk about Mexican immigrants as drug dealers, criminals and rapists, followed by discussion of the deportation of 11 million undocumented immigrants and any American-born children with undocumented parents, most Latino voters changed their minds. The party brand suffered at the hands of its most popular candidate, and now all the Republican candidates hoping to be the next nominee are disadvantaged.

In a 2005 book, “Setting the Agenda,” the political scientists Gary Cox and Mathew McCubbins discussed the importance of a party’s reputation or brand. They were writing about the House of Representatives, but their logic about brands applies nicely to most electoral settings. At the most basic level, the argument is that a strong party brand serves all members of the party seeking election, because a strong signal is more valuable than a weak one. And of course a popular brand is more desirable than an unpopular one. Think of it as quality control.

In less than a year, Mr. Trump has both weakened the party signal and made it less popular, especially among groups that the party needs to court. Many of his rivals were naturally positioned to fight his destruction of the party brand: Marco Rubio and Ted Cruz are both the sons of immigrant parents who fled Cuba; Jeb Bush has said illegal immigration is often an “act of love” to provide for families; and John Kasich has termed himself the party’s “prince of light and hope,” in an effort to distinguish himself from the dark talk of the last few months. But none of these story lines have saved the party from Mr. Trump’s language.