A couple of things: Heavily revamp surveillance, personnel, and other resources at the border (taking that away from the military budget to make it deficit neutral), give amnesty to the illegals already here and register them, expand temporary guest worker programs and in general just streamline and quicken the migration process: migrants are a huge economic plus and are one of the things that keep the US economy young and vibrant unlike the moribund demographics of Europe and Japan.
Several times in the past we've given amnest to illegals already here. You know what happened? Yeah, MORE illegals streamed accross the border to get in on the next amnesty. What do you call someone who does the same thing over and over again and expects different results?
Yes, it has happened in the past - but hey, the US underwent a very strong period of high growth after that, drawing more in, something unlikely to happen again.

Besides, like I said before, high migration is both a symptom of a healthy economy as well as a driver of it.
Sure, you argue that illegals drain public finances. That could be true, but does that outweight other benefits that they provide? If one believes in free market theory, the fact that there are people demanding this labor as well as providing it, voluntarily, would indicate that it is mutually beneficial trade: the basis of capitalism - indeed, free migration flows is part and parcel of a free market, just as capital flows are [and which most people do not complain about].
Not only this, but illegals do jobs that Americans won't do - at that wage. The lower cost of production inputs (in this case labor), in theory, should spur production of whichever industry was utilizing these lower cost resources. Furthermore, illegals are often poor, and the poor have a higher marginal propensity to consume - this means that they spend a proportionally larger part of their income on consumption (even if they are saving for remittances), which is taxed [indeed, one of the taxes that the poor must pay - since illegals are often poorer, they would likely not pay much in income or capital gains taxes, meaning they do not scheme the system as it would appear at first glance].
Lastly, there's remittances, which provide a more indirect bonus; kind of "free foreign aid" not paid by the state but with similar longer-term benefits inherent in helping poorer countries in the region - after all US companies sell many US-made products their goods in these markets . Indeed, Latin America imports proportionally more from the United States than from anywhere else, and maintains a high trade deficit with the US (US sells more products to us, than we sell to the US).
Amnesty doesn't work. Ship all illegal who are caught here back home. Make e-verify mandatory. Fine and jail their employers and their landlords and anyone providing them shelter. Any illegal caught using someone's social security number should do prison time before they are shipped home.
That's a little extreme.
BTW, Japan doesn't exactly take in immigrants...they have a lot of foreign workers...but you can't get into their country and become a Japanese citizen, they don't let you do that. My brother was born there and he's not a Japanese citizen. You actually have to be Japanese to be a Japanese citizen...shocking, isn't it?
Yes, that is why Japan is currently shrinking in population, and why their age pyramid will suffocate their social security system in the coming decades. The shrinkage of Japan is one of that country's most serious long-term problems. Japan is losing around one million people a year despite having one of the highest life expectancies; a similar occurrence is happening in Russia and much of Europe. Some like the (US, Canada, Australia, and Britain) have maintained healthy replacement levels of population growth thanks to open immigration. Again, I have no way to say that immigration does not have any problems, but the effect is very mixed, and should theoretically be beneficial.