This is a many-sided problem. Needed to pin this down are the port, the date and the passenger list itself or secondary information from some indexing source.
Step 1. Hunt for personal papers - a diary or journal, perhaps - or a passport. That would make it easy.
Step 2. Determine an approximate time for the transoceanic trip or trips (since many came for a while, worked, returned home for a bride and then came as a family). Sources for this would include:
Census information from the 1900 and 1910 U. S. censuses and possibly some state censuses which might list the year of entry, years in the U.S., citizenship date. Information may not always be consistent or absolutely correct because of hazy memories or only approximate knowledge by the person supplying the information to the census taker, approximations of dates or miscommunication with the census taker.
Naturalization papers -especially the declaration of intention - should have the date of arrival in the U.S.and port of entry. This date may be exact or an approximation. Sometimes the date of arrival recalled by the individual is when he arrived at the city in which he located rather than the date of arrival at the port!
Death certificates sometimes included the number of years in the area and would give a clue as to the date of immigration.
Obituaries sometimes gave this sort of information - especially in the newspapers of smaller towns.
Step 3. Access indexed information, with a caution as to problems encountered with spelling (you may have to try a dozen or more!), with the number of people with the same name and age, with strange interpretations of handwriting, with hard-to-read copy, with women traveling under their maiden name (and sometimes the listing of the children accompanying the mother didn't give their correct surname), etc.
Microfilm indices are available at the National Archives or through the LDS (Mormon) Family History Centers or sometimes other libraries.
Jun 16,1897-Jun 30, 1902 An alphabetic index of names(115 reels). Jul 1,1902-Dec 31,1943 A soundex index of names (755 reels).
For the pre-1897 era, the gap is being closed with the series of volumes by Glazier and Filby, eds. "Italians to America", which essentially is the transcription of Italian passengers from passenger lists. Each volume is indexed separately. The series started with 1880 and by May 1996 had been completed through 1891. The information, however, does not necessarily seem to be complete!
Step 4. Necessary if the ancestor doesn't appear in any form of index. Consult the passenger lists themselves. Knowledge of the approximate date is essential. This is a slow and tedious process and success is elusive - pages may be missing or damaged or faded, passengers may be listed only by the equivalent of Mr.(surname), etc.
Notes:
(1) The passenger list does not include crew members and some may have been able to sign on as crew to help with the large number of people on shipboard.
(2) Some simplification may be possible by narrowing down logically the ports of entry and embarcation (Example: People from the province of Molise probably left from Naples while those from northern Italy usually left from Genoa, LeHavre or Marseille).
(3) The Morton-Allan directory showing dates of sailings and port arrivals may be useful.
References:
Colletta. John P. "They Came in Ships" Ancestry, Inc., Salt Lake City 1989; revised ed. 1993.
"Morton-Allan Directory of European Passenger Steamship Arrivals at the Port of New York, 1890-1930, and at the Ports of Baltimore, Boston and Phildelphia, 1904-1926" Immigr. Info. Bur., New York, 1931
Glazier, Ira A. and Filby, P. William. "Italians to America. Lists of Passengers Arriving at U.S. Ports, 1880-1899" (partial, 5 vols. completed - through 1891)