Since there is absolutely nothing in US Immigration law that states they must be released while awaiting their asylum hearing (which will probably be denied), if they enter the county illegally...i.e. anywhere other than an official port of entry...they should be given the option of being detained until their hearing or being returned to their country of origin. That may slow the flow.
Additionally, I would like to see the law changed so any employer caught with illegals working for them is be fined a minimum $250,000 per person...fine assessed to the company and the owner (or CEO) jointly and severally with the fines not dis-chargeable in bankruptcy.