...nor deny to any person within its jurisdiction...
Person. Not citizen. They have committed a crime by entering illegally. But if the "state" does not choose to arrest and prosecute...or even if they do...this does not throw out the rest of the Constitution. Besides, they are a 'person' within the state's jurisdiction.
That has been my problem with the 14th for a long time. The word 'person' should have been 'citizen'.