Mash the avocados, sour cream, 3 tablespoons of the lime juice, and the chives in a large bowl. Season the avocado mixture with salt and pepper to taste. Spoon the mixture into a clear glass serving bowl, forming an even layer and smoothing the top.
Squeeze the crabmeat gently to remove any excess liquid, and pick through it to remove any bits of shell or cartilage. Gently mix the crabmeat, mayonnaise, sliced basil, jalapeño, and the remaining 1 tablespoon lime juice in another bowl. Season with salt and pepper to taste. Spoon the crab mixture over the avocado mixture, forming a second even layer and smoothing the top. Cover and refrigerate until ready to serve.
Heat 3 inches of oil in a wok or deep skillet over medium-high heat. Working in batches, add the tortilla wedges and fry, stirring often so that they cook evenly on both sides, for about 3 minutes, or until they are crisp and golden brown.
Using a slotted spoon or a mesh strainer, transfer the corn chips to paper towels to drain any excess oil. While the chips are still hot, sprinkle them lightly with salt.
Garnish the dip with basil leaves, and serve with the tortilla chips.