Interesting theological question I never thought of until reading your quote: Is Paul saying that salvation will be ascribed to the family if the Philippian jailer, individually, professes belief on Lord Jesus Christ?
If you read the next two verses you’ll see that they spoke the Word of the Lord to him and to all who were in his house, they must have become believers, because then the whole family was baptized.
There's doesn't seem to be enough scripture to actually clarify that idea...But here's an interesting verse...
1Co 7:14 For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband: else were your children unclean; but now are they holy.
One thing to consider is the scripture that says two will become one flesh...
One preacher I listen two says that being sanctified is being 'set apart'...The unsaved husband/wife is constantly in a spiritual turmoil...They are no longer part of the world, but not yet saved...