Question: I cannot get my mom to stop bad-mouthing my dad in front of my young children. (My parents have been divorced for years, but the animosity is still very bad.) I have asked nicely, pleaded with her, and resorted to desperation tactics like covering her mouth with my hand when she starts on a rant. But because we all live in the same home, and sometimes they are alone with her, I know I can never completely eliminate the problem.
How do I protect the kids from hearing negative things about their grandpa every single day?
Answer: You can't, not unless you're willing/able to move (her) out.
There is one thing you apparently haven't tried, though, on the silencing-Grandma front, and that is to point out what's in store for her if she keeps this up.
Instead of diminishing Grandpa, she's going to drive the kids away from her, because eventually they will be old enough to make the connection that what she says about one person, she could easily say about them. They can't prove that Grandpa did what Grandma accuses, but they hear with their own ears that Grandma will trash someone, despite being asked to stop.
Not that I expect this to be The Thing that changes her ways; she's probably beyond that. It's just to cover all of your persuasion bases before you move on to this:
Talk to your kids. Explain to them that you love Grandpa, you think he's a good person, though not perfect of course, and Grandma is angry at him and won't change her mind. Say it enough times so that you can cue them when Grandma starts ranting - an "Ooh, there goes Grandma again"-type comment that will pretty well neutralize whatever your mom is saying.
The kids will make up their own minds - and, too, they're going to ask questions as they get older. Be ready to explain, on a fair, respectful, yet kid-friendly level, why Grandma is so angry and why you've chosen not to respond to him the same way.
This will also give you an opening, when they're old enough, to talk about the futility of holding on to anger versus the pragmatic value of forgiveness.