I believe Pope Benedict stated that the fundamental solution would be to eliminate the need to immigrate.
People emigrate from Mexico because
(1) We have a better business environment because of less corruption than Mexico.
(2) our better-educated population has the productivity to be able to afford to supply better social services for the poor.
We could reduce the incentive of Mexicans to move here by reducing our level of social services to what it is in the lowest Third World counties (zero). Putting the whole world on our welfare system is not workable -- we're broke as it is.
Fixing Mexico would involve invading them and killing off their corrupt ruling class -- I don't think the Left would support that.