1) Require that all immigrants, tourists and visitors be bonded for some substantial period. They would upon entry to the US have to purchase an insurance policy that would indemnify the public in the event of any criminal activity. This would place the burden upon the company or individual posting the bond to figure out who the high risk candidates were. One advantage of this approach is that it would help weed out folks that perpetrate the more common crimes like drunken driving. Penalties for evading bonds—or profiting from that type of illegal immigration-would have to be high enough to pay the risks imposed by illegal immigration on the public.
2) Screen potential immigrants using a process similar to a jury system. If 12 US citizens agree that someone is a questionable candidate after meeting with him for a couple hours, then just maybe, that person isn`t a good fit for life in the US. Exceptions might be made for folks of exceptional talent(and with the exceptional eccentricities that sometimes accompany such talent).
3) Peter Brimelow has previously suggested that the US should strive towards a very low level of net immigration. The simple fact is that US immigration rights have tremendous economic value and if there are folks in the US that don`t really identify with the culture in the US, there are folks ready and willing to take their place-and many of those people would have the means to provide considerable incentive to trade places with the existing US citizen or permanent resident. Steve Sailer has suggested a "buyout" of Muslims in France might be appropriate. If these incentives were permitted here, I would suggest the most alienated of US citizens would be among the first to take advantage of them.
There are serious constitutional questions around screening candidates along religious lines. I think we have a lot of other steps that could be taken first.