Well, one historical problem with large scale immigration in the United States was the notion that immigrants were taking American jobs. If you look back into the early 1900's, when New York had a large number of Western European immigrants coming into the city, many business owners would put up signs like, "Help Wanted - Italians Need Not Apply", and this was interchangeable for any group business owners did not want applying for open jobs.
This claim is not entirely false though; the Center for Immigration Studies (CIS) has done research into immigration's effects on employment in the United States. By adding immigrants to the workforce, the U.S. GDP (gross domestic product) increases by $1.7-2 trillion a year. They have concluded that Americans hit by the brunt of this influx of immigrant workers are ones who are less-educated, most likely seeking blue-collar jobs, where immigrant workers would work for less than Americans.
Source: https://cis.org/Testimony/Impact-LargeScale-Immigration-American-Workers
Furthermore, if you are discussing illegal immigration, a massive issue is the literal disappearance of illegal immigrants once they enter the United States. As we can see from border issues today, the infrastructure cannot handle the sheer numbers of immigrants coming in, and once immigrants are caught, registered, and released into the country, it is very difficult to track them moving forward. This also encourages human trafficking over soft borders; it is estimated that drug cartels make over $150+ billion a year trafficking women and children over the U.S.-Mexico border.
