Yes and No. A sovereign nation should certainly protect it's borders. And the USA is a sovereign nation. In the US during the early 1900's and the industrial evolution there really was no immigration policy. We largely had open borders. However, we also did not have a massive set of entitlement programs. When my grandparents came from Sicily, there was no Medicaid, supplemental income or any other program. The incentive to come was for economic freedom, which they had never really had.
Today is a bit different. Far too many people who come here are immediately placed on social entitlements which are already on a fiscally unsustainable path. The assimilation process into American culture is not expected as it once was. My grandfather insisted that only English was spoken in his house at the dinner table. He was an American first. And he wanted his kids to raised as Americans, with American culture. He was an Italian American, my father became an American Italian. Today...I am just an American. Today too many groups have 2nd and 3rd generations that have not embraced American culture and still barely speak English. The other issue is also national security threats, which we are all intimately familiar with.
Immigration is good for the country when it is done in a controlled manner, and we can keep track of who is coming into the country, what skill sets they offer and what their intentions may be. If you don't have a degree of immigration, and you couple that with an aging population...you can run into many of the economic problems Japan is experiencing at the moment due to their poor demographic trends. But this must be done in concert with both national security concerns and entitlement reforms. Although I have no confidence that with be the case anytime soon.
But to assume we can't take everyone in is somewhat misleading. We can...as long as they are coming here to be economically productive self sufficient citizens, in which case they are more than paying for themselves. And I think that most people come here with that intent. The Kaufman index for Entrepreneurship shows foreigners are 40% more likely to start their own business. The problem isn't necessarily the people who are coming here as much as it is the political figures who look to undermine their ambition, dump them on social entitlements & make them wards of the state for a guaranteed vote. Which leads to unproductive behavior and generations of gov't dependency.