This issue comes up from time to time:
Can a person convert to Judaism if they are married to a non-Jewish spouse who is not interested in converting with them?
The answer is generally "no" for Orthodox conversion, "yes" for Reform conversion, and "maybe" for Conservative conversion.
Here is an official opinion written in 1993 that was accepted by the Committee on Jewish Laws and Standards of the Rabbinical Assembly (of the Conservative Movement):
The Case of the Unconverted SpouseThe above opinion ruled that it is acceptable to convert a person to Judaism if they are married to a non-Jewish spouse. However it is clear from the vote of 8 in favor, 11 against, and 4 abstaining, that it is not accepted by all Conservative rabbis. Note that there are many reasons why rabbis may vote against or abstain and without a dissenting opinion, you can't really know what about the ruling they disagreed with or had reservations about.
The result is that some Conservative rabbis will absolutely not supervise conversions for people married to non-Jewish spouses, but some will, and I suspect for the latter case, it will depend somewhat on the situation. For example, I do not think many rabbis will want to do a conversion if the spouse objects or is not supportive. I happen to know of a couple in my minyan in which the wife converted under Conservative auspices early in their marriage along with their first child who was a baby, but the husband did not because although he supported his wife's conversion, he was still not sure of his own religious feelings. Some years later, the husband did convert.