Where do I begin? I know my great-grandfather, Josef Boksa, 2/28/77 was born in Warsaw. And recently found his naturalization papers from 1917. It lists the date of his arrival as 8/16/1902 and that's all I know. He lived in Chicago. I've not been successful finding any of the ship records for my family.
You could attack this problem from either the port of departure or the place of entry to the USA.
If you choose to begin in Europe, then you should look for
- Passenger Lists leaving UK 1890-1960 (FindMyPast has this)
- Hamburg passenger lists 1850-1934 (available on Ancestry)
Since it is more likely that migrants from Poland would travel through Hamburg than travel first to England, I would start with the German lists.
There is a small chance that Josef sailed for America from Bremen, LeHavre, Amsterdam, Rotterdam, and Antwerp. In that case, your chances of success are much less.
If you want to begin at Josef's destination, then you will want to hope that he came through Ellis Island but be aware that there there are several other east coast ports that might be the one.
As an alternative strategy, set aside the name of the person and focus on the date of arrival (and hope it is correct). Search major newspapers from mid-August 1902 for reports of migrant ships and when you have a list of names begin to investigate where they came from and the ethnicity of their cargo. That may narrow the possibilities.
The 1920 Census shows that the (deceased) mother of Josef's children was born in Poland. Do you know if they met in the USA or came here as a couple. Often a family group is easier to track than an individual.
There's a likely suspect in the Hamburg passenger lists, departing 22 Sept 1901 from Hamburg and arriving at Boulogne-sur-Mer; Plymouth; New York. It lists his DOB as "about 1873" on the Ancestry indexing but the handwritten (in German) passenger log is next to impossible to read so dates are always a bit... iffy... in those cases.
Hope that tidbit might help.