Actually, difficult immigration is very poor economic policy, and that the only justifications anyone ever comes up for it are invariably jingoism, racism, or cultural conservativism only goes to illustrate that. We want more people -- greater manpower, at any level of education, means a more productive economy; more people working -> greater outpot -> larger economy -> people making more money -> greater output -> larger economy, ad infinitum.
This is Economics 101 stuff, folks. Every developed and developing economy has been growing steadily and without cessation, for decades, if not centuries, since the advent of the various institutions that enable market economies to function. (And, for the anarcho-capitalists in the room, that includes government -- market economies require government rules and regulation to function properly.)
Singapore, a newly developed economy, makes it very easy (and not even on a relative scale, but even on an absolute scale) for noncitizens to enter the country and work, and, if they choose, to become permanent residents. Singapore consists mostly of one city built on the country's primary island, with a land area of 699 square kilometers, or 270 square miles, give or take (land reclamation changes the acreage) and a population of 4.6 million, for a population density of 16 thousand people per square mile. Compare to New York City; the five boroughs have a collective land area of about 300 square miles and a population of 8.2 million, making for a density of about 27 thousand per square mile.
Now, while Singapore is far from utopia, the leadership there generally makes quite sound economic decisions (I do say generally; they have made some decisions that are *questionable* at best), and it has a strong and, most notably, resilient economy. They also enforce immigration very strictly; if you're there illegally, or if you hire illegal immigrants, you WILL be up **** creek. But the substantial number of expatriates and guest workers (people who go there from much poorer countries to work for a while, send money home, and eventually return home to buy the farm and raise a family) does not drain the economy -- quite the opposite, in fact -- nor has it ever done anything to compromise things that are uniquely Singaporean in that country's vibrant culture.
Compare to Japan, which is facing a negative birth rate, an aging population that is facing an ever-growing problem of just how they're going to be supported by the steadily shrinking population of young people, and an economy that has been stagnant since the early 1990s and shows no signs of renewed life despite the Japanese government's best efforts to reinvigorate the economy; the single best thing that Japan could possibly do? Open the country up to immigration. Not just legally, but socially; Japan is deeply xenophobic, and even when they're not racist towards gaijin (most Japanese, for instance, have brutally racist views of Brazilians, similar to some of the racist rhetoric spewed about Hispanic immigrants in this country, only worse) they don't ever believe anyone who isn't Japanese could ever understand what it is to be Japanese. Immigrants from less developed economies could enter the country and work menial jobs that Japanese people are not inclined to take (Brazilian people do this sometimes, which is why they're vulnerable to that kind of racism, but they're not really permitted to attempt to assimilate, let alone find try to gain permanent residency or citizenship); educated people from developed economies could enter the country to contribute to the country's corporate and academic health.
The US currently stands somewhere between the two; Canada's slightly closer to Japan with its immigration policy, but still pretty open, but most European countries are varying degrees of hostile to immigrants, both legally and socially. The French are experiencing a frightening amount of social unrest because of the deep divide between the white, nominally-Catholic French majority and the nominally-Muslim, black Algerian-descended minority; Switzerland makes it all but impossible for any foreigner to do anything but retire to the country.
Of these countries (and I was just picking some more or less at random that I happen to know something about), the American economy is by far the strongest and the largest.
Arguing against an open-door immigration policy is, frankly, stupid. It's shortsighted and perhaps even racist; every economy, developed or otherwise, only benefits from having more people participating in it on any income or education level. There's even significant economic advantages and motivations for assimilation, especially on a generational level, so arguing that immigrants are going to somehow destroy American culture is, frankly, preposterous.