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Heat 2 tablespoons of the oil in a large deep skillet over medium-high. Pat shrimp dry with paper towels. Season shrimp with ½ teaspoon of the salt, and add to skillet. Cook, stirring often, until shrimp are cooked through, about 3 minutes. Transfer to a plate; set aside.
Heat remaining 4 tablespoons oil in same skillet over medium-high. Add garlic, and cook, stirring often, until aromatic, about 1 minute. Add tomatoes, crushed red pepper, and remaining 1 ½ teaspoons salt. Cover and cook, stirring occasionally, until tomatoes are tender and release their juices, about 14 minutes.
Meanwhile, bring a large pot of salted water to a boil over high. Cook spaghetti until al dente. Drain spaghetti, reserving 1 cup pasta water.
Add shrimp and any accumulated juices to tomatoes, and cook over medium. Add cooked spaghetti, Parmesan, and ½ cup reserved pasta water, tossing to coat. Cook, stirring occasionally, until sauce clings to pasta. Remove from heat. Garnish with fresh basil, if desired. Serve with additional Parmesan, if desired.