We propose to extend laser-cooling techniques, so far only achieved for neutral molecules, to molecular anions. A detailed computational study is performed for $C_{\bar{2}}$ molecules stored in Penning traps using GPU based Monte Carlo simulations. Two cooling schemes—Doppler laser cooling and photodetachment cooling—are investigated. The sympathetic cooling of antiprotons is studied for the Doppler cooling scheme, where it is shown that cooling of antiprotons to subKelvin temperatures could becomes feasible, with impacts on the field of antimatter physics. The presented cooling schemes also have applications for the generation of cold, negatively charged particle sources and for the sympathetic cooling of other molecular anions.