I think that the previous posts were quite strong. The idea of what determines "illegal" immigration is probably where the most amount of intellectual clash will be present. When immigrants did come to America, it seemed to me that it was more readily accepted that they would come here. The debate came about as to whether or not they should be here, and not whether or not it was illegal. The current debate about immigration seems to rest much on the issue of its legality as more of a cover for the discussion about the presence of immigrants in America. Certainly, there has to be a legally sanctioned way to ensure to which immigration in America is to be adhered. Yet, I think that the discussion has become so politicized and so politically charged that one has a difficult time in assessing the truth of the issue. I believe that America's history has always been one where the battle between Nativists who wish to stop or curtail immigration meets against the inexorable pull that America has for immigrants to it. This ebb and flow seems to be a part of America's history and its narrative.
One way to look at this issue is the historical fact that we have never in our history since 1619 had a functional economy without some kind of either captive or exploited labor force. The 236 year history of slavery in this country certainly has a strong measure or influence over our development and prosperity in the US. Immigrant populations in the 19th and 20th centuries took the most difficult, dangerous and lowest paying jobs, creating huge profits for corporations and individuals. In the modern day, we are somewhat schizophrenic on this issue, if you'll pardon the expression. That is, we want the low prices on our food, and we want to pay no more than minimum age in the service industries, at the same time as we want our borders secured and all illegal immigrants to leave. We have created both an economy and a society that depends on a steady stream of immigrants, both legal and illegal.
I am not quite sure what you are asking here, but I will try to answer.
To me, the most important connection between illegal immigration today and US history is with the two previous big waves of immigration (1840s and late 1800s). In both of these previous waves of immigration (which were legal because there were no immigration laws back then) many of the same issues arose in the minds of the "native" Americans.
In both waves, Americans thought that the new immigrants were going to hurt the US. They thought the immigrants were not going to be able to assimilate and become "real" Americans. They thought that immigrants had bad values and contributed to increased crime rates. These are issues similar to the ones today (except for the issue of the fact that current immigrants are *illegal*).
When I look at history, these are the events that seem to me to have the most connection to the current issue of illegal immigration.