Bring a large pot of water to a boil. Heat the oil in a very large skillet set over medium heat. Add the onion and bell pepper and cook 4 to 5 minutes, or until tender.
Add the linguini to the boiling water and cook until tender, about 7 to 8 minutes.
Add the wine to the onion and bell pepper, bring to a simmer, and reduce it by half. Stir in the cream, return to a simmer, and simmer 2 minutes. Stir in the pesto, salt and pepper. Add the crab claws to the cream and heat them through, about 2 to 3 minutes.
When the pasta is cooked, save 1/2 cup of the cooking liquid. Drain the pasta well and toss it, and the 1/2 cup of cooking liquid, into the cream sauce. Garnish servings with basil sprigs and, if desired, serve with parmesan cheese for grating at the table.