In West Virginia, Donald Trump won by over forty percent of the vote. Hillary Clinton got little more than a quarter of the vote.
Prior to the late breaking allegations of sexual assault, they actually polled on exactly this question, as reported by Real Clear Politics. At that time, the difference between Joe Manchin voting for Brett Kavanaugh and voting against him was twenty-seven percent. That's roughly the same as Hillary Clinton's vote share in West Virginia in 2016 (perhaps even a little higher, although different significant digits).
Of course, as you note, Democrats may not bother to vote for him. The allegations presumably changed the political calculus somewhat. It's unclear though what happens now or what could have happened if he had voted differently.
- We don't know if every Hillary Clinton voter will not vote for him as a result of his voting against the Christine Blasey Ford position.
- We don't know if the allegations changed how the people whose vote depended on his Kavanaugh vote felt. Perhaps if the identical poll were taken now, it would show different results.
What we do know is that prior to the allegations, even with the potential for Hillary Clinton voters to not vote for him if he voted for Kavanaugh, his margin improved by twenty-seven percent. Voting for Kavanaugh was as important as the entire Hillary Clinton vote. And that's assuming that none of the Hillary Clinton voters had already dropped out if he made that choice. Remember, Kavanaugh was unpopular before the allegations. After the allegations, his popularity only moved about five percent (although his net popularity moved about ten, since it counts both sides).
Assuming that means that Manchin will lose five percent as a result of Democrats abandoning him. And that the effect of voting for Kavanaugh has been reduced by five percent above and beyond that, Manchin still nets seventeen percent over voting against Kavanaugh.
Until a new poll is done, we won't have good data on this. We can only speculate on how things may have changed since July.
It's worth noting that it is quite possible that he did do a poll on that question. Manchin may know what happens (at least in a poll) if he had voted for or against Kavanaugh.
Even with a new poll, there's still time for things to change between now and election day. New evidence could be made public that either exonerates Kavanaugh or proves that he lied (perjury is certainly an impeachable offense). Manchin, who had an opportunity to view the private release of the FBI report, may have reason to expect such evidence to be on the exonerate side.
Finally, even if Democrats find Manchin inadequate, how many would prefer Patrick Morrisey? Elizabeth Warren and Kristen Gillibrand are not running in West Virginia. Their options are Manchin or Morrisey. They may hold their nose and still vote for Manchin even if disgusted by his Kavanaugh vote. Or this may be the last straw. We need more data to build a case either way.