A fluid droplet suspended in an extensional flow of moderate intensity may break into pieces, depending on the amplitude of the initial droplet deformation. In subcritical uniaxial extensional flow the nonbreaking base state is linearly stable, implying that only a finite-amplitude perturbation can trigger breakup. Consequently, the stable base solution is surrounded by its finite basin of attraction. The basin boundary, which separates initial droplet shapes returning to the nonbreaking base state from those becoming unstable and breaking up, is characterized using edge tracking techniques. We numerically construct the edge state, a dynamically unstable equilibrium whose stable manifold forms the basin boundary. The edge state equilibrium controls if the droplet breaks and selects a unique path towards breakup. This path physically corresponds to the well-known end-pinching mechanism. Our results thereby rationalize the dynamics observed experimentally [H. A. Stone and L. G. Leal, J. Fluid Mech. 206, 223 (1989)].