On Sunday, Department of Health and Human Services officials will open a shelter for up to 1,000 minors at Lackland Air Force Base in Texas, authorities said, and will begin transferring youths there by land and air. The level-four alert is the highest for agencies handling children crossing the border illegally, and allows Homeland Security officials to call on emergency resources from other agencies, officials said.

In an interview on Friday, Mr. Johnson said the influx of unaccompanied youths had “zoomed to the top of my agenda” after his encounters at the McAllen Border Patrol station with small children, one of whom was 3.

The children are coming primarily from El Salvador, Guatemala and Honduras, making the perilous journey north through Mexico to Texas without parents or close adult relatives. Last weekend alone, more than 1,000 unaccompanied youths were being held at overflowing border stations in South Texas, officials said.

The flow of child migrants has been building since 2011, when 4,059 unaccompanied youths were apprehended by border agents. Last year more than 21,000 minors were caught, and Border Patrol officials had said they were expecting more than 60,000 this year. But that projection has already been exceeded.