Here is the English translation of the fields in the Hamburger Passagierlisten.
1. Surname (family members are grouped together using brackets)
4. Previous residence (may or may not be birthplace)
5. State or Province
8. Number of people
9. Children under 10
10. Children over 10
11. Children under 1 year
Lately I’ve had some great luck using the Hamburger Passagierlisten, 1850-1934 aka the Hamburg Passenger Lists, 1850-1934 at Ancestry. This is a departure (emigration) database, rather than the arrival database for New York. In typical fashion, the Germans were more thorough about completing forms, so I have found more of the all-important information about hometowns / birthplaces in these emigration records than I’ve found in the corresponding immigration arrival records in New York.
Ancestry helpfully notes that since the records are in German, it helps to search in German, and only the years 1877-1914 have been indexed at this point. They also suggest browsing the Handwritten Indexes, 1855-1934 if your ancestors have not yet been included in the index.