I would say that the threat of terrorism and "terrorists" themselves would be one particular group of which Americans are fearful. The attacks of September 11 are still a fear in Americans' minds. Airports, for example, are a constant reminder of the threat of terrorism. Issues of border control are immediately fed into the fear of terrorism. From this fear, there are other groups that might be fed into this. There still is a belief that Islam is closely associated with terrorism, and while this is categorically invalid, it is still there. The lack of control over border security can also be fed into the fear of terrorism. Justified or not, those fears are present and are quite real in American society today.
It's not clear from your question if you are simply asking for groups that might be targeted during a witch hunt (I noticed that you tagged your question with The Red Scare), or as the first answer suggests, you are asking about illegal immigrants. Since the first answer discusses illegal immigrants, I'll offer a couple of other groups.
Obviously, after 9/11 and even in recent times, many Americans fear people who look Middle Eastern. Some have suggested ethnic profiling at airports and unlimited questioning based purely on someone's appearance or religious adherence.
Another group that is being portrayed as a group to fear (I'm not sure how many Americans actually fear them) is the Tea Party group. They are often portrayed negatively by the media as violent and socially backward. While I am not a member of such a group, I was downtown in my city when a tea party rally occurred last April. I wasn't frightened--it seemed to be a large group of people from all age ranges at a typical political rally.
One last group that has garnered a lot of attention and fear from many circles is the government itself. With political and economic uncertainty, I have heard more and more of my students discuss their fear of the government. Whether this is rational or not is one's opinion, but my students and others fear financial burdens in the future and a hopelessness about being able to find jobs even if they go to college.
The news of recent years seems to indicate that many Americans are fearful of powerful congressmen. In his new book, In Search of Self-Governance, Scott Rasmussen observes that most Americans
have come to believe that the political system is broken, that most politicians are corrupt, and that neither major political party has the answers.
Rasmussen adds that the gap between the people who want to govern themselves and the politicians who want to rule them may be as large today as the gap between the colonies and England in the 1700s.
In a Fox News survey of just this month (March,2010), American voters expressed deep anxiety about the fragility of the economy of the United States as they face uncertainty about jobs and "the public sector spending." Only 30% expressed confidence in the size and role of the government, while 65% (05% were noncommital) say that the government is now "too big" and "is restricting personal freedoms."
With the passage of The Patriot Act and the controversial issues about the health care bill, there are people who feel that their fears are reasonable about "personal freedoms." Certainly, their anxieties about the economy seem fairly reasonable. Unemployment is very high in the majority of states, schools have been forced to close in cities such as Kansas City, MO, and Detroit, MI. Some states in the U.S. are even insolvent; California's financial situation is dire. The National Debt is "astronomical" and people worry that Congress and the President will not, and cannot, fix these problems.
From your question, you are implying that illegal immigrants are a group that Americans fear today (because you ask about what rights legal immigrants should have). I think that you are right that many Americans do fear illegal immigrants and their influence on the country.
Is this reasonable? This is totally a question of your opinion.
You can argue that it is not a reasonable fear because America has swallowed up many immigrant groups before without being harmed.
However, you can argue that this time there are too many immigrants who come from a nearby country and therefore will not really want to assimilate and become American.