Heat the vegetable oil in a large pot over medium heat.
Add the chopped onion and cook until translucent, about 5 minutes.
Add the minced garlic and ginger, and cook for another minute.
Stir in the red curry paste and cook for 2 minutes.
Pour in the coconut milk and chicken broth, and bring to a simmer.
Add the mixed vegetables and chicken pieces, and cook until the chicken is cooked through and the vegetables are tender, about 15 minutes.
Stir in the fish sauce, brown sugar, and lime juice. Adjust seasoning to taste.
Remove from heat and stir in the fresh basil leaves.
Serve hot with steamed rice.