Ms. Harris’s remarks on Friday in the border town of Douglas, Ariz., laid out a vision that makes clear that her party — and the nation — continue to back away from the long-held American promise of protection to desperate people fleeing poverty and violence abroad no matter how they enter the United States.

“The United States is a sovereign nation, and I believe we have a duty to set rules at our border and to enforce them,” Ms. Harris said. “I take that responsibility very seriously.”