The traditional right of sanctuary -- that a person can flee from the justice of the state (or Christian barbarians sacking the city, as in the case of the Vandal sack of Rome) to the mercy of the Church -- is part of Western Civilization. Of course, properly implemented, people seeking sanctuary are obliged to stay in the church to which they fled, live on the charity the church provides, and devote their lives to prayer, at least until the circumstances from which they sought sanctuary abate.
I have no problem with churches continuing to offer sanctuary in that sense, and would argue that it would violate the First Amendment to prevent them from doing so, in light of the long-establishment of the practice in Christian tradition. On the other hand, picking up the odd illegal alien who is in a "safe house" rather than a church, would not trample on the traditional Christian practice.