In a covered 5- to 6-quart nonstick pan over high heat, bring 2 1/2 to 3 quarts water to a boil.
Meanwhile, rinse green beans; trim and discard ends, then cut beans into 2- to 3-inch lengths. Rinse chicken and pat dry; thinly slice crosswise into 1/8-inch-thick strips.
Stir fettuccine and beans into boiling water; cook, uncovered, until pasta is barely tender to bite, 2 to 3 minutes. Pour into a colander, rinse with hot water, and drain.
Rinse pan and set over medium heat. Stir in 1/4 cup coconut milk and curry paste. Add chicken; stir until no longer pink in the center (cut to test), about 3 minutes. Add remaining coconut milk, 1/4 cup water, sugar, vinegar, and 1 tablespoon fish sauce; mix.
Add pasta and beans to pan; mix until noodles are hot, about 2 minutes. Add more fish sauce if desired. Pour into a wide bowl. Sprinkle with basil leaves.