Use the word musty to describe an unclean, stale, and possibly moldy smell. If something smells musty that means it’s probably been sitting in a damp corner without any air for a little too long.
Musty smells take time to develop: they’re caused by dampness and mold, and signal some amount of decay. You might also call the smell of a musty old basement rank or foul. But people also use musty to insult someone’s writing, or their ideas. A musty speech wouldn’t just be old-fashioned, it would be old in a bad way: rotten and boring.