Is your beagle spayed? Did the vet mention anything about spay incontinence? It's a condition in spayed dogs that results in a weak bladder. They usually don't even notice they're dribbling and most often you notice it in the morning when they get up and you see the puddle.
If that's what it is, it can be treated with estrogens or with phenylpropanolamine quite successfully.
"We are--each of us--dying; it's how we live in the meantime that makes the difference."
"It's not what you gather, but what you scatter that tells what kind of life you have lived!"
"Be kinder than necessary, for everyone you meet is fighting some kind of battle."