In a food processor, pulse basil and butter together until thoroughly combined and basil is in tiny bits. With a rubber spatula, scrape the basil butter into a small dish. Refrigerate until ready to use.
Heat a charcoal or gas grill until hot. Lightly brush grill with oil. Grill corn about 10 minutes, rotating frequently, until tender and lightly charred.
Serve hot, with cold basil butter, and sprinkled with salt.