Place a large pot of salted water over high heat to boil.
Drop the corn into the boiling water and cook until bright yellow and tender, 5 minutes. Remove the corn to a plate and let cool. Drop the pasta into the boiling water and cook to al dente according to package directions.
While the pasta is cooking, add the tomatoes, crab, and basil to a large mixing bowl. When the corn is cool enough to handle, cut the kernels off from the cobs and add them to the bowl.
When the pasta is ready, drain it well and add it to the mixing bowl with the vegetables and crab. Drizzle in the oil, season with salt and pepper, and toss well to combine. Serve the pasta warm or at room temperature garnished with the cheese.
This pasta is great served up as a cold salad for lunch or a BBQ the next day. The pasta tends to absorb any liquid or oil after resting overnight, so if you're going to serve it the next day, toss in a few drizzles of olive oil to bring the dish back to life.