Crispus is mentioned here and in 1 Corinthians 1.14.

The word represented by family may be expanded to mean “people of his household.”

The verb tenses translated heard and believed suggest that the persons who did so heard and believed over a period of time rather than at a single moment. The Greek has no expressed object of the verb heard, though most translators supply either the “message” (Phillips) or “Paul” (Revised Standard Version, An American Translation*). Some translators, however, supply no object but simply have “listened and believed” (New English Bible; see also Moffatt).

The passive expression were baptized may be rendered active as “Paul baptized them.” Note there is a specific reference to the baptism of Crispus in 1 Corinthians 1.14, and though the family of Crispus is not specifically mentioned, there is a parallel reference in 1 Corinthians 1.16 to Stephanas and his family. Rather than making Paul the agent of the baptizing, it may be possible to use what might be called a substitute passive by saying “they received baptism,” “they underwent baptism,” or “they experienced the rite of baptism.”

Quoted with permission from Newman, Barclay M. and Nida, Eugene A. A Handbook on The Acts of the Apostles. (UBS Handbook Series). New York: UBS, 1972. For this and other handbooks for translators see here .