In the early 1800's port cities in the USA bore the burden of immigration. By the time they arrived, so many immigrants were tired, hungry and poor they ended up in the City Almshouse. This meant the citizens had to take care of them. At first the citizens of the city asked the Mayors for funds to support the poor. Eventually they asked the states, and by mid-century some states (Pennsylvania, New York, Massachusetts) set up State agencies to deal with the issue. Eventually, beginning in the 1880's, the Federal Government nationalized the programs.
Dating back to the colonial era, New York City assumed responsibility for its citizens who were destitute, sick, homeless, or otherwise unable to care for themselves. The city maintained an almshouse, various hospitals, and a workhouse on Blackwell's Island (now called Roosevelt Island) for the poor.
These Almshouse records often contain immigration details, such as name of ship, date of arrival in USA and port of arrival.
Olive Tree Genealogy has an ongoing project to transcribed and publish New York Almshouse Records. The first set is for the years 1819-1840 and includes Ship Captain's Name, Date of Bond, Sureties, Date Discharged, Death Date, Remarks, etc.
For example, under date 1820 March 11 Elizabeth Kennedy age 34 is listed as having died June 14, 1820; her daughter Mary Ann died Nov. 5, 1820
Researchers can use the clues in the Almshouse records (admission date, ship captain's name, owner's name, etc) as well as census records, to narrow the time frame of arrival. Families with children born in one country, such as England, and then in New York will find it much easier to narrow the time frame of immigration.
For individuals recorded in 1855-1858 Almshouse Records for New York City the information includes ship name, date of sailing, ports of departure and arrival