You're getting close answers. From 1820 - 1920, Castle CLINTON operated on the lower tip of Manhattan. It was still in operation even after Ellis Island opened. The reason is that it was a smaller facility and handled everyone who came over first and second class, while the steerage passengers got off at Ellis Island and were processed in the cattle pen.
Technically, everyone entering through Castle Clinton came through "The Port of New York". It's the predecessor to the Port Authority and was operated by the State of New York. Ellis Island was a federal facility and its records are completely separate.
Castle Clinton is now in the care of the Battery Conservancy. They are in the process of transcribing all of the records, but admittedly with more than 90 years of records to process, it will take a few more years. I can find about half of my immigrant ancestors on there. Irish records from the Potato Famine are a major part of the hold up...so many of you had ancestors fleeing by the thousands everyday. It's going well though and you can search the site at
http://www.castlegarden.org
The other places they may have entered are the ports of Boston and Philadelphia. Some of those records are searchable through HeritageQuest and Ancestry.com (free if you go to the library to search them...you have to pay for a subscription if you do it on your own). The films are also available through your local library, the National Archives and the LDS Family History Centers.
1888 is a decent year to search. The vast majority of those records have survived. But unless you know the actual ship's name and date of arrival, you may have to wait for them to be transcribed.
Good luck and happy hunting.