Tucked away in southern New England, Danbury, Connecticut, is far from the U.S. border with Mexico, but illegal immigration is very much a local problem. Residents complain about overcrowded houses, linguistic balkanization in schools, and wage depression. The mayor has called for state police to receive special immigration-enforcement training to contend with the influx. By some estimates, the blue-collar city of 77,000 now has an illegal population of around 15,000. Many have moved in over the past decade to take landscaping and construction jobs in affluent neighboring suburbs but not without placing a strain on the community’s infrastructure. In April,...