During April and May, federal authorities separated at least 1,995 children from parents facing criminal prosecution for unlawfully crossing the border, the Trump administration announced on Friday. Photographs and an audio tape of crying children caught in the administration’s dragnet stoked new outrage this week, leading members of both parties to express concern.

Mr. Trump and two members of his cabinet have forcefully defended the “zero tolerance” policy, with the president repeatedly making the false accusation that only the Democrats, the minority party, were to blame for it. The law does not require the authorities to separate children.

The issue, which turns on federal immigration policy, might have been largely a question left to leaders in Washington, not governors. But the Trump administration’s request for National Guard troops earlier this year has thrust the matter into the state-level debate as well.

President Trump called in April for the National Guard to be deployed to the border, saying that thousands of troops were needed to stanch illegal crossings, even though they are at a 46-year low.

In recent days, Democratic governors from states like Colorado, Delaware, New York and Rhode Island have expressed deep opposition to family separation, and many declared they would not help at the border. For many of the governors, the defiance — which was being embraced by left-leaning constituents in their states — was largely symbolic, since they had not been planning to send large numbers of soldiers in the first place, if they had been asked at all.

“Under normal circumstances, we wouldn’t hesitate to answer the call,” said Gov. John Carney, Democrat of Delaware, who said he had declined a request on Tuesday to send troops to the border. “But given what we know about the policies currently in effect at the border, I can’t in good conscience send Delawareans to help with that mission.”