Lay 2 prosciutto slices on a flat work surface so that they overlap slightly. Scatter 4 basil leaves on top. Add 1 barramundi fillet; wrap up. Repeat with remaining prosciutto, basil leaves, and barramundi.
Heat olive oil in a large skillet over medium heat. Add prosciutto-wrapped barramundi; cook until flesh flakes easily with a fork and prosciutto is slightly crisp, 3 to 5 minutes per side.