In large bowl combine corn muffin crumbs, mayo, mustard, egg, red pepper, red onion, basil, salt and pepper. Gently fold in crabmeat. Shape mixture into 4 to 5 crab cakes or whatever size you like. Chill crab cakes for 1 hour.
Heat 1-2 Tbsp. olive oil or butter in medium skillet. Saute the crab cakes over medium high heat for 3 minutes per side until golden brown. Do not turn the crab cakes before 3 minutes so a nice crust forms. Serve with garlic aioli.
Garlic Aioli: Combine all ingredients in medium bowl and whisk together. Chill. Serve with crab cakes.