They could, but they won't. The 'Pubs have two Silver Hammers this election season with which to beat the 'Rats into a pulp, if they only have the guts and smarts to use them. First, energy independence (drill here, drill now). The second, no taxpayer-funded benefits for illegal immigrants (note illegal immigrants, the 'Rats always lie/recast the debate in terms of immigration, which no one is against as long as it's is legal). Those are sure-fire winners, ripe for the taking. The 'Rats have staked out public positions on these issues that are at odds with a significant majority of public opinion, and they will have a hard time either selling those to the public or backpedaling their way out of them (i.e., "nuanced").
But the key to success is that the 'Pubs have to successfully exploit the issue and be ready for the inevitable 'Rat response of deflection and obfuscation. They can't let them get away with that. They have to keep focused on the two plain, simple issues. If they do that, they win. If not, no one will care, or, worse, they'll let the 'Rats blame it on the 'Pubs (which the media will enable).
Tourists also have to add any children they have had born in this country to their "family visa" just as soon as they get it.
That means none of those tourists' kids have ever been considered citizens under US law. After all, US citizens don't need a visa to be here.
Quite possibly the single most cogent analysis of the 2008 Presidential election I've seen.
Too bad the GOP won't do it.