Remittances are huge, hence the suggestion to halt/tax them was such a blow to Mexico: millions of households suddenly unable to pay rent/utilities -- it'd be a cascading nightmare.
So the answer is to let X% of the population in so 5X% can receive foreign money but still be subjected to a broken environment?
One word; remittances. They already make up far more money than foreign aid, and the best part is that it's far harder for corrupt governments to get their hands on it. Don't underestimate the passion those few people who enter the country has for their families back home.