Refugees are not "illegals". They have the legal right to come into the USA and claim refugee status.
The crisis at the border is the treatment of refugees by the Trump Administration. Before Trump started separating families, and holding refugees in internment camps, there was no crisis.
Then what is an Illegal?
An illegal crosses the border, and disappears into the night, with no intention to return to his homeland.
Refugees must be standing on US soil to claim asylum. That's the law. If they cross the border illegally to enter, it's a misdemeanor which will be dismissed if they're granted asylum.
Trump has closed the ports of entry, and only allowing a handful of asylum seekers to enter each day. It's called "metering". He's hoping to trigger riots, or refugees rushing the border - anything to prove the refugees are dangerous and shouldn't be allowed in. Plus he gets to count them as "illegal entries", and thus inflate his illegal numbers to make things look worse than they are.