Whenever you have truth it must be given with love, or the message and the messenger will be rejected. ~ Mahatma Gandhi
Liberals and conservatives have enough conflict without inventing more. It's hard to compromise. What makes it harder is when either side mischaracterizes the position of the other and makes fun of the extreme characterization as if it were what was really at issue.
Every country with the means does something to define its borders and regulate who comes in and out and who stays. U.S. immigration laws have procedures for people wishing to become citizens, and we prioritize based on who we'd like to see come in.
Conservatives see anchor babies as a circumvention of our immigration laws. By an application of the 14th amendment that wasn't an issue when it was passed, non-citizens who could not themselves become citizens can force the U.S. to accept their children as citizens by giving birth here. Then, because their children are entitled to rights as citizens, the parents get to raise the children here, citizens or not. After 18 years, the child is grown, but it doesn't seem right to deport the parents who have built a life here. Thus, the child is the "anchor" for the family.
Sensible conservatives (yes, they do exist, though maybe not in Congress) are not afraid that these children will grow into terror babies. Maybe some of the tinfoil hat crowd is, but as far as I can see, sensible conservatives don't like 1) seeing this back door access to life in the U.S. exploited 2)by people who could not get in the front door. They view this back door as letting in unskilled workers who have come from poor areas and are probably bringing communicable health problems with them.
Is our immigration system a good system? Not really. Is it due for an overhaul? Most definitely. This is something on which we can all agree.
Work from that starting point by at least acknowledging their real concerns. Data would be great. How many are coming in this way? What % is this of total undocumented residents? Where do these parents settle? Do they get jobs? Pay taxes? Carry diseases? This is what conservatives (like my parents) want to know. Some conservatives are incurably racist, but some are just going by stereotypes and are willing to be educated. We owe it to them and to ourselves to address their concerns and break the stereotype.
My point is, if you want to work something out that conservative voters can support, don't ridicule them for being afraid of babies. Conservatives will assume--and not unreasonably--that you'd rather ridicule them than discuss actual issues, and that you're trying to change the subject because you like the open back door. Like the term "death panels," the term "terror babies" shows you're really not here to debate the merits, you're here to stir up emotions (and insult the voters).