What was the goal of many nativists with respect to immigration?
Nativism, appeared in the political arena during the 1840's under The American Party, otherwise referred to as the 'Know Nothings'. The party stood on an anti-Catholic platform, especially those immigrants from Ireland. In 1856, at the height of their political influence in America, the American Party sought to restrict immigration prohibiting immigrants from holding any government office, even if they were naturalized citizens. In addition, the terrorist group Ku Klux Klan perpetuated hate by preying on the fear of what the newcomer might incur on those already here. Usually centered on competition with regard to jobs, economics, and political power, these groups that perpetuated 'Xenophobia', the fear of foreigners finally manifest in Congressional legislation beginning in 1882 with the Chinese Exclusion Act followed by The Gentlemen's Agreement in 1907, the Literacy Test Act in 1917, the Emergency Quota Act 1921 and again in 1924.
You do not specify which period in American (I assume) history you are asking about. My answer will address the period from the late 1800s to around the 1920s. In other words, the second major wave of immigration to the US.
During this time, many nativists wanted to restrict immigration to the "right" kind of people. This meant that they wanted only white people and only certain kinds of white people. They wanted Northern and Western Europeans as opposed to Southern and Eastern Europeans.
This desire was fulfilled in the 1920s with laws that were passed to restrict immigration and favor the Northern and Western Europeans.