Many claim that because the massive immigration that occurred a century or so ago worked out OK, the current massive immigration will similarly work out OK.

This argument is an instance of the logical fallacy "Appeal to Tradition". The context has changed from a century ago. While there are certainly parallels, there are many differences and it's quite a leap to claim that the current situation will work out the same.

In fact, here are just nine of the differences between then and now:

1. Many of today's immigrants are from Mexico, and the U.S. Southwest was briefly Mexican territory. Most past sending countries never held territory inside the U.S. The belief that the Southwest was stolen from Mexico is widespread in that country; in fact, a Zogby poll conducted in Mexico had 58% saying that the U.S. Southwest rightfully belongs to Mexico. There's also an indigenous movement holding that anyone of "native" blood should be able to travel anywhere within the Americas at will. There's no elite opposition to those sentiments, and in some cases the elite supports those sentiments.
2. Many of our current illegal aliens are from a neighboring country, meaning they don't have to make a clean break, they can go back and forth. There are families with members on both sides of the border as well as a few million mixed-status families inside the U.S. (i.e., families with some legal/citizen members and others as illegal aliens).
3. Related to that, past immigrants came here on ships; current immigrants can and do walk over.
4. There wasn't a far-left, Gramscian "multiculturalism" movement a century ago. The related issue of political correctness makes it difficult for some to, for instance, use the correct names for things ("illegal aliens") rather than euphemisms ("undocumented workers").
5. There were ethnic newspapers, but nothing like today's ethnic media.
6. Immigrants who came through Ellis Island were checked for disease and suitability. And, they were pre-screened by the cruiseship companies, who were charged if someone was rejected. Nowadays, anyone can overstay their visa or just walk across.
7. There's been a rapid increase in dual citizenship, leading to U.S. citizens with divided loyalties. 14% of U.S. citizens are eligible to be dual citizens, and Mexico encourages dual citizenship as a way of obtaining political power inside the U.S.
8. The welfare state hardly existed a century ago.
9. Obvious to anyone who's been to, say, Dallas or Los Angeles, there were many fewer people here a century ago than there are now.

Note that Univision anchor Jorge Ramos agrees with me on at least part of the above. In 2008, he said ( peekURL.com/z8xwFdB ):

The Hispanic community has created its own cultural space within the United States. That's different than for any other group of immigrants in the past. ... It's easier for someone in New York or Chicago to talk to the family via the Internet or a cell phone, or simply to go back and forth to Mexico or El Salvador, than it was for an Italian immigrant to go from New York to Sicily.