Per usual, the libertarian approach would solve the problems on all sides:
1) Substantial reductions in the welfare state would send all the leeches back where they came from, leaving only truly hard-working immigrants in place.
2) Elimination of the FICA/Medicare mandate would likewise eliminate the institutional advantages illegals have re. wage demands in building/service trades (self-employed U.S. citizens currently have a 15% disadvantage)
Our government forces some to labor for the benefit of others. Eliminate that injustice, and you mitigate the vast majority of hostility towards immigrants.