I think we're about to see some of the more arcane rules of debate in the Senate take center stage: repeated quorum calls, filibuster, cloture, etc. I suspect this will not go down without a fight.
Filibuster was abolished for SCOTUS nominations.
There was nothing procedurally that Democrats could have done to stop Kavanaugh. If there were, they would have.
They cannot stop this nomination by traditional means.
A walkout? That wouldn't do anything. There's 53 sitting Republicans. That's enough to confirm a nomination.
What they should do is physically barricade the Capitol. I’m not really joking. It will come to that.
My first guess of the most action they will take:
Staging a walk-out of the vote.