Place dried anchovies in a large pot and add cold water; let soak for 20 minutes. Bring liquid to a boil; reduce heat to medium and simmer for 20 minutes. Remove anchovies.
Wash bean sprouts under cold water 3 or 4 times, discarding any bad or brown heads of bean sprouts. Wash cabbage and cut into bite-size pieces.
Cook cabbage in a pot of boiling water for 5 minutes; rinse under cold water and squeeze out excess liquid.
Mix bean sprouts and cabbage in the anchovy broth and add soy bean paste, stirring until soy bean paste dissolves. Add garlic, red pepper powder, and salt; cook and stir over medium-high heat for 15 minutes.
Stir green beans and dashi into soup, lower heat to medium, and cook for 15 minutes more.