Cook rice according to package instructions. Meanwhile, place cornstarch in a small bowl; whisk in 2 to 3 tablespoons chicken broth until smooth (reserve remaining broth). Set aside.
Heat oil in a large skillet over medium. Add green beans and onion; cook, tossing frequently, until they begin to soften, 5 to 7 minutes.
Add curry paste to green beans and onion in skillet. Cook, stirring and scraping bottom of skillet, until paste is slightly darkened, 3 to 4 minutes. Add chicken, coconut milk, cornstarch mixture, and reserved chicken broth. Simmer until vegetables are tender and sauce is slightly thickened, 5 to 8 minutes.
Add basil and lime juice. Season with salt, and stir to combine; serve over rice.