American Indians are the most anti-immigrant (that's the Indians with the feathers, by the way): Using General Social Survey data, I calculated a mean score for each ethnic group in America on a question if one agrees that immigrants' ideas and culture improve American society. Answers ranged from "strongly agree" (1) to "strongly disagree" (5) so a high mean indicates that the group is least likely to think that immigrants make things better.
Mean anti-immigration score
American Indian 2.76
The focus on immigration these days is from Mexico, so it is no surprise the poor minority groups see little benefit from having tens of thousands of low-skill immigrants come to the U.S. each year to compete with them for jobs. Black liberalism offsets this a bit; American Indians are not as liberal. White ethnics seem less likely to buy the hype as well. Ethnic solidarity seems to trump job competition for Mexican Americans.