We investigate the transport of active polymer chains in steady laminar flows in the presence of thermal noise and an external constant force. In the model, the polymer chain is worm-like and is propelled by active forces along its tangent vectors. Compared with inertial Brownian particles, active polymer chains in steady laminar flows exhibit richer movement patterns due to their specific spatial structures. The simulation results show that the velocity-force relation is strongly dependent on the system parameters such as the chain length, bending rigidity, active force and so on. The polymer chain may move in some preferential movement directions and exhibits absolute negative mobility within appropriate parameter regimes, i.e., the polymer chain can move in a direction opposite to the external constant force. In particular, we can observe giant negative mobility in a broad range of parameter regimes.