Position an oven rack about 4 inches below the broiler and heat the broiler on high. Arrange the prosciutto in a single layer on a small baking sheet and broil until it begins to curl, 1 to 2 minutes. Flip the prosciutto and broil until it appears dry-crisp and has curled a bit more, about 1 minute. Let cool, then finely chop or crumble by hand; set aside.
Slice the kernels off the corn cobs for a total of 3 cups corn. Reserve the cobs.
In a medium Dutch oven over medium heat, melt the butter. Add the onion and cook until softened and slightly golden, 5 to 7 minutes. Season with a generous pinch of salt.
Add 4 cups of water, the broth, potatoes, 1-1/2 cups of the corn, the cobs, and 2 tsp. salt. Bring to a boil. Reduce the heat to medium low and simmer until the potatoes are tender, 10 to 15 minutes. Remove from the heat and discard the cobs.
Working in batches, carefully purée the soup in a blender, transferring each batch to a large heat-proof bowl or large liquid measuring cup.
Pour the puréed soup back into the pot. Add the remaining 1-1/2 cups corn and bring to a boil over medium-high heat. Reduce heat to medium low and simmer, stirring occasionally, until the corn kernels are tender, 3 to 5 minutes. Season to taste with salt and pepper. Garnish each serving with the crisped prosciutto and basil.