Combine chicken broth, water, onion, jalapeno, ginger, and garlic in a large pot; bring to a boil. Add chicken breasts; cook until no longer pink in the center and an instant-read thermometer inserted into the center reads at least 165 degrees F (74 degrees C), 15 to 20 minutes.
Remove chicken from the pot. Cool until easily handled, about 5 minutes. Shred chicken and return to the pot. Stir in bok choy and cook until wilted, 3 to 5 minutes. Season soup with salt.