I would argue that Americans really did not have to “cope with” immigrants in times past. Americans who were born in the US have always been the dominant political, economic, and social forces in our society. Immigrants have had to cope with their new environment, but the natives of the US have not really had to cope with the immigrants.
In general, the US has dealt with immigrants in two ways. First, they have used the immigrants to provide relatively cheap labor, often for unpleasant jobs. Immigrants to the US provided much of the workforce that built the transcontinental railroads. They provided many of the workers who toiled in the factories that made the US rich during the late 1800s and early 1900s. The owners of American industry were able to use the immigrants to help build their businesses. Americans in general benefitted as the US economy grew.
Second, many Americans have dealt with immigrants through nativism. This means that Americans have often harbored negative attitudes towards immigrants. Americans have felt that immigrants take jobs from natives of the country. They have felt that immigrants are more likely to bring crime, political radicalism, and (particularly in past times) disease. They have blamed immigrants for many social problems. This has been the cause of nativist movements like that of the “Know-Nothings” in the 1850s, the movement to expel the Chinese in the 1880s, and the KKK in the 1920s.
To cope with something is to deal with the difficulties or problems that it brings you. Immigrants in historical times did not really cause problems that Americans had to cope with. Instead, they helped the American economy grow. Some Americans felt that immigrants were bringing problems to the country and they “coped with” those problems through nativist movements.