A practical coplanar Josephson junction left-handed transmission line based on a step-edge junction technique is proposed in high-temperature superconductor technology and analyzed by electromagnetic simulations. The layout is designed for monolayer Yttrium Barium Copper Oxide thin film fabrication process. The propagation stopbands are tunable by controlling bias currents of Josephson junction arrays acting as parallel inductors for unit cells of the transmission line. Unlike the reported designs, each unit cell of our left-handed transmission line is independently biasable due to dc isolation of the unit cells along the transmission line. Being individually biasable is practically important for cancelling out junction critical current nonuniformities originated from the fabrication process. Tunability of the onset frequency of the left-handed propagation regime, and the propagation stopband separating the left- and right-handed propagation regimes of the proposed device has been demonstrated by simulations, is in good agreement with theory, and is predicted to be above 15%.