Immigrants to the U.S. who do not speak English well tend to earn less. And their children might suffer, too. That's the conclusion of some work summarized in the NY Times by Austan Goolsbee:
By comparing the outcomes of English-speaking and non-English-speaking immigrants who arrived in the United States around the critical period age, they [Bleakley and Chin] document that poor English skills meant less schooling and substantially lower wages for immigrants and that these disadvantages often extended to their children, even if those children were born in the United States.The Bleakley-Chin study is available here.
One of their simplest demonstrations of this fact compares immigrants from different islands of the Caribbean. They document that the wages and education levels of immigrants from non-English speaking islands like the Dominican Republic or Puerto Rico look similar to those of immigrants from English-speaking islands like Jamaica and Trinidad — as long as the person originally came to the United States by age 11. For those who were older when they arrived, however, immigrants from non-English-speaking islands do significantly worse, on average, than those from English-speaking ones. Non-English speakers are much more likely to drop out of school and also have significantly lower-paying jobs when working. Their finding points strongly toward language as the deciding factor, since the differences exist only after age 11.
... In turns out that children whose immigrant parents came to the United States when young do just about the same in school regardless of whether the parents came from English-speaking or non-English-speaking countries. But the situation is different for children whose parents were older when they arrived. The children from non-English-speaking households do much worse than English-speaking ones. They are less likely to go to preschool and much more likely to drop out of high school.
When Professor Bleakley and Professor Chin compare the overall distribution of test scores of English- and non-English-speaking families, they find that the big differences appear mainly among children with the lowest performance. The top half of students from non-English-speaking households do just about as well as the top half from English-speaking households. It seems that a child with talent can succeed no matter what the parents' skills are, as has been true for centuries in this country. But parents whose English is poor have a big negative impact on the below-average children.