Actually, I think the issues are much much different in Europe where in fact that have closed borders and much much more immigation enforcement than the US. The issues that led them to here are:
1. A generous welfare state that took away incentives, including the incentive to have kids to take care of one in ones dotage.
2. Immigration as a solution to a labor shortage and to pay for the welfare state.
3. A religious decline that implies the Euros don't understand religious people particularly non Christians.
4. A failure to adopt a guest worker or assimilation program.
Unfortunately some anti immigrant Americans want to glom on to a completely different situation.
Afterall America may have an enforcement problem but we are talking about yet another group of Christian immigrants that America has absorbed for generations. Certainly in some locals where the left is in charge they are trying to stop assimilation, but elsewhere the US situation is completely different than the Euro situation of a population of unassimlated but legal immigrants or their offspring.
I have never set foot in Europe proper, it sounds like you may have or have studied these issues in more depth than I.
In your comments you refer to the welfare state or cheap labor being some of the key reasons so many have migrated to the continent. Sadly , I don't think that is that far from applying here as well in some areas of this country. Of course , that does not speak very well of the countries of origin from whence the immigrants originate.
Re: closed borders, I do not know what procedures are in place to prevent any ethnic group or immigrant(s) from buying a Eurail pass or taking boats in the dark of night and moving freely thru and between the continent's many countries and acting out their own nefarious deeds without needing to circumvent reasonable preventive measures.
I am taken back a bit by your comment about anti immigration activists "glomming" on to what you describe as a different situation for the most part. I am not so sure that is necessarily true, or whether you harbor some feelings that there has been an over-reaction by the same to our current plight here.
Here we have more than an enforcement problem, imo, we also have a leadership problem and an awareness one as well, albeit intentional or otherwise.
I would like to think that if nothing else, many here have been working within the system to change some of the sentiments that keep us bound to failed policies.
Maybe you could be a little more specific where glom comes in. Or maybe I am just seeing gloom where none exists. ;-)