Place chicken in the crockpot.
In a bowl, whisk together coconut milk, curry paste, chicken broth, garlic, ginger, fish sauce, and sugar.
Pour sauce over chicken in the crockpot.
Add onion, bell pepper, and green beans.
Cover and cook on LOW for 4–6 hours or HIGH for 2½–3 hours.
Stir gently and shred chicken if desired.
Add lime juice and adjust seasoning.
Garnish with fresh basil or cilantro before serving.