I would not use Listerine in a dog's mouth. It is strong, will sting and they will swallow it.
First, you should try to determine the cause of the bad breath. The most likely cause is dental issues, plaque, gum disease, etc.
foul smelling breath can also be a sign of a systemic disorder such as diabetes, liver or kidney problems.
There are specific products, such as toothpaste, mouth sprays, dental chews, that are made specifically for dogs.
Give the dig's mouth a good inspection, to start with to check for plaque, red, irritated gums, etc. pay particular attention to the back molars.
Your dog may need a good Vet exam to determine the cause.
I think that's a very bad idea. Maybe try a couple drops of lemon juice. It kills bacteria.