To be clear Europe as it is structured with the way it thinks about it self now and for a number of decades is a long term threat to the existence of the United States of America. The solution is easy and solves most problems, we just don't do it.
First this has nothing to do with their Islam problems or the European Union. Really it doesn't have much to do specifically with its economic tail chasing(Greece), lose of shared values (replacing religion) or stunning military decline. Those are symptoms.
Their problem, argue it all you want, is social democracy. It is a slow socialism that erodes the individual. It is so far down the road that the lack drive to pass one self (your wisdom, views, ect.) to the next generation is leading to France not being French. If not France pick the country. Mark Steyn and a few others understand and explain not only the birth rate problem but the problem on the whole.
Why is this a problem for the US? Their collapse is inevitable, long term.
Cliche as it is, you do run out of other peoples or countries money. Likely a country will get a weak government or messiah when they collapse. Then the weak government is overthrown by a tyrant or the messiah become one. I can not imagine either blaming the idea a social democracy and embracing the solution of individual liberty and hard work.
The answer to varying degrees is to blame the other and the other won't the in the tangled dying mess in Europe. China? It may be weaker than Europe when it succumbs. The US may not be much better but it has a history that can be used much more effectively.
The underlying solution to most foreign policy problems to be absolute evangelists for liberty in every thing we do. We must sell not democracy or freedom but individual liberty. Why it is moral, how it strengthens the individual, how it solve practical and economic problems has to be talked about clearly to our enemies and especially our friends. He is a clearer explanation if I remember correctly.
It needs to come from our government in general but the military and state department specifically. Just as important is the education on campuses. As if its not bad enough that we send foreign students back with technical skill that we need here is, we send them back with at lest disdain if not hated for liberty. Does liberty even get a chance on to defend itself on most campuses? If it doesn't then how can it in Europe?