LOS ANGELES — The Trump administration announced Monday that it is dramatically stepping up prosecutions of those who illegally cross the Southwest border, ramping up a “zero tolerance” policy intended to deter new migrants with the threat of jail sentences and separating immigrant children from their parents.

“If you cross the Southwest border unlawfully, then we will prosecute you. It’s that simple,” Attorney General Jeff Sessions said in announcing a policy that will impose potential criminal penalties on border crossers who previously faced mainly civil deportation proceedings — and in the process, force the separation of families crossing the border for months or longer.

The new policy could flood the immigration courts, already suffering severe backlogs, and create new detention space shortages for federal agencies that even now have been forced to release many undocumented immigrants until their cases can be heard. Mr. Sessions said he has dispatched 35 additional prosecutors and 18 immigration judges to the Southwest border region to help handle expanding caseloads.

The stepped-up enforcement strategy marks the Trump administration’s toughest move yet to stem the flow of migrants into the United States, though officials said the category of migrants accounting for much of the recent surge, those seeking asylum from violence in Central America, will still be able to apply for legal refuge.