Unrestricted immigration would reduce global income inequality. I don't judge the worth of individuals by their nationality, so it is global inequality that I must be concerned about, if income inequality is my concern.
In the US, average wage for employed individuals is about 42000 [2]. Since there are 300 million Americans and 6.7 billion non-Americans, America should shift almost entirely to the average to get income equality. This will mean the equivalent of a 55% across-the-board pay cut for every American, provided prices do not go down. Your rent will stay the same, but your income should go down 55%.
(This is in fact happening, but is manifesting as uneven inflation without corresponding wage rises, so you'd experience an 81% rise in average prices, rent, food, restaurants, ..., while seeing your wage stay the same)
Also note that average worldwide unemployment is at least a factor of 3 higher than in the US. So it would not just take a massive pay cut, but also firing close to 20% of all workers, to bring the US into the worldwide average.
This is why socialists are (or should be) against immigration, and should be pro-isolationist. I think that if you want to improve the conditions of the American working class, that would be the correct attitude to have.
In reality the period of 1950-today has been a very exceptional period in history where not only incomes were higher, by most measures, than ever before, but income inequality was far lower than the vast majority of economic history.
[1] http://www.bbc.com/news/magazine-17512040 [2] http://www.worldsalaries.org/usa.shtml
I think it's better to try to help people in their own country, so their local economy is improved instead of allowing just anyone to emigrate to your own country.
One of the ironies of income-redistribution schemes. People are happy to soak the 1% to pay for benefits for our lower/middle-classes, but if poorer people come along it's a strain, never mind that they almost certainly need it more.
As for helping people in their own country -- it is unfortunate but (barring conquest) there's only so much you can do in places ruled by kleptocrats, rife with corruption, and with only weak protections of the rule of law.