“The most important thing is, the president has to bring this crisis to a humane and resolute end — quickly,” said Simon Rosenberg, founder of the New Democrat Network, a think tank that has focused on the growing importance of Hispanic voters. “If you want immigration reform to pass soon — whatever ‘soon’ is — it is crucial that the government’s response to this crisis goes well, and the president has to do everything he can.”

Rep. Raúl M. Grijalva of Arizona, another Democrat who has called on Obama to visit the Texas border to view the situation firsthand, said the way the president handles the border crisis will leave an imprint on his legacy among the Hispanic voters who were so important to his reelection and remain a key to the future of the Democratic Party.

Latinos blame Republicans for the failure of comprehensive reform, he said, but now they want to “see who’s standing with us.”