The Greek word here is ἁγιάζω, which the KJV translates as "sanctify" or "make holy", but more modern versions (e.g. RSV) sometimes translate as "consecrate". It is in this sense, for example, that the word is used sometimes in the Septuagint, such as:
Numbers 6:11–12 LXX
καὶ ποιήσει ὁ ἱερεὺς μίαν περὶ ἁμαρτίας καὶ μίαν εἰς ὁλοκαύτωμα, καὶ
ἐξιλάσεται περὶ αὐτοῦ ὁ ἱερεὺς περὶ ὧν ἥμαρτεν περὶ τῆς ψυχῆς καὶ
ἁγιάσει τὴν κεφαλὴν αὐτοῦ ἐν ἐκείνῃ τῇ ἡμέρᾳ, ἧ ἡγιάσθη κυρίῳ τὰς
ἡμέρας τῆς εὐχῆς, καὶ προσάξει ἀμνὸν ἐνιαύσιον εἰς πλημμέλειαν, καὶ αἱ
ἡμέραι αἱ πρότεραι ἄλογοι ἔσονται, ὅτι ἐμιάνθη κεφαλὴ εὐχῆς αὐτοῦ.
And the priest shall offer one for a sin-offering; and the other for a whole-burnt-offering; and the priest shall make atonement for him in
the things wherein he sinned respecting the dead body, and he shall
sanctify his head in that day, in which he was consecrated to the
Lord, all the days of his vow; and he shall bring a lamb of a year old
for a trespass-offering; and the former days shall not be reckoned,
because the head of his vow was polluted.1
I believe this latter sense is the way we are to understand 1 Corinthians 7:14. This is also the sense that it seems to have been understood in antiquity. John Chrysostom writes, for example,
"The unbelieving husband is sanctified through his wife, and the
unbelieving wife is sanctified through her husband." Yet earlier he
said, "Do you not know that he who joins himself to a prostitute
becomes one body with her?"2 We might then conclude that a
woman joined to an unbelieving idolater becomes one body with him. Is
there a contradiction here? No, because although they become one body,
the woman is not defiled, since the purity of her faith is stronger
than the impurity of his unbelief. Likewise the purity of a believing
husband is stronger than the impurity of a believing wife ... There is
hope that the spouse who is perishing through his unbelief might be
saved through marriage to a believer.3
The historical context is also relevant, I think. At the time that Paul was writing, Christians were a very small minority. Cases abounded where one or other of a married pagan couple would accept Christ before the other. Paul is addressing a practical marital issue here, in guiding spouses how they are to continue to relate to their still unbelieving spouse. The advice given here shows that Paul felt it was better for a couple to remain married, with the believing spouse living in the hope of his or her spouse eventually converting, than for the believer to simply abandon the unbelieving spouse.
1 Brenton English translation
2 1 Corinthians 6:16
3 Homily XIX on 1 Corinthians, tr. from Greek in St. John Chrysostom: On Marriage and Family, p.31