The effect of atom losses on a homogeneous one-dimensional Bose gas lying within the quasi-condensate regime is investigated using a Monte Carlo wavefunction approach. The evolution of the system is calculated, conditioned by the loss sequence, namely the times of individual losses and the position of the removed atoms. We describe the gas within the linearized Bogoliubov approach. For each mode, we find that, for a given quantum trajectory, the state of the system converges towards a coherent state, {\it i.e.} the ground state, displaced in phase space. Provided losses are recorded with a temporal and spatially resolved detector, we show that quantum feedback can be implemented and cooling to the ground state of one or several modes can be realized.