Family is usually considered singular. So the correct word is "lives." He lives.
If you wanted it to be plural, you'd say something like, "Every member of his family live in New York."
1. His family live in New York.
2. His family lives in New York.
[Which one is grammatical? Do we have to use a singular verb or a plural verb?]
2 answers
Singular:
His family lives in New York. (The subject is "family" and thus needs the singular verb form.)
Every member of his family lives in New York. (The subject is "member" and thus needs the singular verb form.)
Plural:
All members of his family live in New York. (The subject is "members" and thus needs the plural verb form.)
His family lives in New York. (The subject is "family" and thus needs the singular verb form.)
Every member of his family lives in New York. (The subject is "member" and thus needs the singular verb form.)
Plural:
All members of his family live in New York. (The subject is "members" and thus needs the plural verb form.)