You don't need an incentive to get them to bear children, any more than you need artificial inducements to get college students to drink beer. Changing the citizenship rule would have little or no effect on the fertility of illegal immigrants.
There is one category of foreigners who do travel to this country just to give their babies the gift of American citizenship. "Birth tourists" reportedly are catered to by travel agencies, hospitals and hotels offering deals for expectant mothers. One Turkish-owned hotel in Manhattan offers a package including month-long accommodations for $45,000, which doesn't cover hospital costs.
But it's hard to see why this phenomenon causes so much anger among anti-immigrant groups. They complain about poor Mexicans sneaking in illegally, taking jobs, getting government assistance and staying forever. Shouldn't they be happy to hear about rich Turks getting visas, avoiding employment, paying their own costs and leaving immediately?
At any rate, it's not exactly a raging epidemic. The National Center for Health Statistics says only 7,670 babies were born in 2006 to women who said they don't live here, or a microscopic 0.17 percent of all live births.
If we want to head them off, we don't have to amend the Constitution. We can just deny tourist visas to visibly pregnant women, or make it a federal offense to come here solely to have a baby.
But what's the fun in addressing a modest problem with a minor change? Anti-immigration zealots would much rather mount a heroic expedition to conquer a mighty mountain. Even if it's really a molehill.