That view is embraced by most progressives, or at least a more moderate version that wants much more immigration than we currently have. But what's contradictory is that those some people also want a generous welfare state.
Here's Paul Krugman on that very issue:
Democrats are torn individually (a state I share). On one side, they favor helping those in need, which inclines them to look sympathetically on immigrants; plus they’re relatively open to a multicultural, multiracial society. I know that when I look at today’s Mexicans and Central Americans, they seem to me fundamentally the same as my grandparents seeking a better life in America.
On the other side, however, open immigration can’t coexist with a strong social safety net; if you’re going to assure health care and a decent income to everyone, you can’t make that offer global.
So Democrats have mixed feelings about immigration; in fact, it’s an agonizing issue.
My concern here is that this is in fact not an agonizing issue for Democratic voters, while it is most likely on the radar for Democratic politicians. To often, I see rank and file Democratic voters speaking about the legend of infinite wealth, where the government should be strengthening the social safety net with no consideration on costs because America is "rich."
Perhaps I'm wrong, maybe this is something they really do agonize over in private and don't like to talk about in public.
What I find most frustrating about this is the way immigrants are often portrayed as a burden when they should really be considered an asset. Bryan Caplan's analysis of the data tells us that most studies show immigrants pay more in taxes than they use in benefits, and the ones that show a net loss only show a small one.
Our borders need to be open for everyone, even the uneducated and the poor. Maybe we'd make some progress on this issue and pass immigration reform is everyone stopped talking about poor people as victims in need of saving and instead as untapped resources that can help us if given better incentives.