There is a long history of xenophobia and racism in this country, that has come up time and again, and parties have catered to it. The GOP, for example, was founded in large part because slavery destroyed the old whig party (the fight over it), disaffected whigs who were against abolition moved over to the Democratic party (The whigs before this time were interestingly, very similar to the GOP of today, they had the southern planter class, they had some strength in the farm belt, and they had the well of industrialists). There were the nativists reacting to the first waves of immigration, that were catered to by political forces.
The south went to the democratic party for the obvious reason of the civil war, and especially reconstruction, so yes, the KKK and such could be associated with the Democrats, and it why for almost 100 years the south was solidly Democratic. The Democrats had a coalition, that was centered in big city machine politics,they also had their gentry (think FDR), and then they had what Mencken called the “KKK branch of the democratic party”, which included the southern dixiecrats, and also included the farm belt populists and what today we would call the religious right (think of Williams Jennings Bryan, who was both). Like what usually happens, shifts happened because of major events, when the civil rights era started, you had Truman integrating the armed forces (interestingly, the Supreme Court shift was done by Eisenhower, that ruled in Brown vs Ed and later civil rights rulings, and Eisenhower used troops to enforce segregation orders), and then of course the civil rights laws of the 60’s and some things swapped. The old farm populist/southern block moved over to the GOP, while some of the traditional republicans in areas like the Northeast and California shifted to the Democrats (or independent, but tend to vote Democrat).
On the other hand, the history doesn’t really matter, it doesn’t take a genius to figure out who is catering to the nativist/xenophic/racist contingent. It has been primary GOP politicians who have been yangering about being ‘overwhelmed’ by illegal immigrants, it has primarily been GOP politicians stirring up fear of “Sharia Law” and the fear that all Muslims are terrorists, it is an old tradition, fear mongering to get votes, and it is getting traction.Donald Trump comes out and basically calls Mexican immigrants criminals and rapists, and his poll numbers go up, he talks about building a ‘giant wall’ and his numbers soar. Fear mongering goes both ways, there are Democratic candidates who will say that any GOP candidate wants to turn the country into a Christian theocracy, which isn’t fair to those who aren’t like that (for whatever he was, Romney was not planning to do that, nor would Kasich or Bush; on the other hand, the Ted Cruz’s of the world, Huckabee, etc, not an invalid fear). The GOP has used fear and scapegoating to get votes, in the 70’s they told white, blue collar workers that their jobs were being lost to affirmative action hiring of women and minorities (rather than the truth, that those jobs were fleeing to cheaper labor markets, like down south, or increasingly, overseas).
As far as Shariah law goes, what often is brought up is in civil law, not criminal or anything else. It is true that Islam as a religion is very different than Christianity or Judaism or other faiths, in Islam the political and the religious is not seperated, Islamic law has very strict rules about things that are not in other faiths (or at least followed), in terms of things like how loans happen, how divorce happens and so forth. Often, what is called “Shariah Law” has been attempts to allow for example, in contract law, the option (note the word “Option”), for two Muslims to draw up a contract under Islamic principles and have it be binding, they would choose it, if it were between a muslim and a non muslim it wuld only be valid if the non muslim agreed to it. For those who see this as being fearful, financial firms already often Shariah compliant loans and services, to cater to the Muslim community, and it doesn’t change how you or I deal with the same institution. Yes, there have been morons, there was a judge in NJ who in a divorce case ruled that the woman claiming grounds of spousal abuse were invalid, because under the law from where they came, which was some variant on Islamic Law, it was not considered abuse, so therefore he could not grant a divorce based on those standards…and it was totally reversed on appeal and the appelate judges in no uncertain terms told the judge he was an idiot (in legalese), that their home law had no bearing in the US, and shouldn’t have. If a Muslim couple gets divorced in the US, Islamic law has nothing to do with it, it happens under the terms of US law. If they want their divorce recognized in their religion, that is a different matter, but in civil law, it would have no meaning.
The other problem, as Donna pointed out, is that there is no one 'Shariah Law", every country, every sect of Islam, has differing rules, and given that we have Muslims from all over the world, whose would rule? One thing for all those afraid of Muslims wanting to impose Sharia law, I work among more than a few people who are Muslim, who are Shia and Sunni and Sufi, and one of the things the fearful set forgets is that when people immigrate, for the most part, they do so to get away from what they experienced at home, the sectarian violence, being the Sunni minority in a Shia country, being Sufi (whom both Sunnis and Shiites think are blasphemous), they immigrated to get away from that, so why would they want “Shariah” law imposed, when they experienced the horrible face of that from where they came from? The other thing is, you are seeing first generation immigrants, Muslim immigration is relatively recent, and when it hits the next generation it is no different than other groups, the next generations are not as observant and even less likely to want that kind of crap.