That's been my favourite approach all along. Look at the war on drugs- we've tried to stop drugs by targeting the supply, and that's worked smashingly, hasn't it? Trying to target the supply of illegals is just as ineffective.
Now, if the average employer (including white middle-class America, who hire illegal nannies and maids) knew they were going to be hit with a $10,000 fine for every illegal they employ, the economics of hiring illegals would change overnight. Cut off the demand, and the supply goes away.
You do that, you'll start to see what you're now seeing in Muslim neighborhoods in the US- a gradual attrition of the population as it becomes too much of a hassle to remain here.