How might the growing opposition to immigration lead to problems in the United States?
Anti-immigration rhetoric and demagoguery can breed unrest.
The major point of contention (as far as immigration is concerned) in the United States stems from undocumented (some people refer to it as "illegal") immigration. This primarily pertains to people that cross the southern U.S.-Mexico border without going through the "proper legal channels" (North American countries have closed borders (essentially if you're not from the U.S. you can't work in the U.S. legally without going through a lengthy [a year or more] process) unlike the European Union [any citizen of the E.U. can work in any E.U. nation at will]). The rhetoric of the anti-immigration camp in the United States reflects that of Fascist state (namely Hitler in Germany and Mussolini in Italy) party leaders of the early 20th century (and this rhetoric can be imbued with violent undertones). I won't point fingers or mention names but recently in the United States we've been in the process of selecting candidates for the presidential election and as a result this rhetoric has sprung forth to national attention (search "I'll build a wall and make Mexico pay for it" on Google).
In the United States there are two camps, those who support the rhetoric and those who oppose the rhetoric, both of these groups have very strong feelings with respect to their positions. When one group gathers (usually the camp that supports the rhetoric) another group (typically the opposition) protests (initially peacefully). Tensions and emotions run high and literally anything (but usually hateful speech) can cause things to reach a flash-point, a mob mentality sets in and peaceful protests oftentimes turn into mini-riots (or scuffles between individuals).
Most of the people that support the rhetoric state that their ire (anger) lay in the fact that immigrants take jobs away from Americans. In all actuality however the jobs that a vast majority of immigrants (both documented and undocumented) take are at or below the minimum wage (well below what it costs to sustain oneself) and are jobs that are unfathomably horrible (a janitor at a slaughterhouse for example [or, as another example, one my great-great grandfather's, who was from Italy, had a job putting adhesive glue on envelopes]).
Personally, I believe that the anti-immigration rhetoric isn't the view of the populous and that it is something that people use to lay blame to their own personal situations (essentially it's something they can target their anger at rather than blaming the true reason for their circumstances, themselves).
I honestly think (and hope) that this is just an idea held by a minority political fringe-group, that more people have latched onto this anti-American idea because of their inability (or unwillingness) to engage in some serious introspective thinking, or that they have a genuine ignorance regarding matters of immigration, a sense of disenfranchisement (by the people who actually believe these ideas), or, worst of all, in an attempt to thwart boredom.
I hope this helps!