The design and characterization of a new laser-desorption molecular beam source, tailored for use in x-ray free-electron laser and ultrashort-pulse laser imaging experiments, is presented. It consists of a single mechanical unit containing all source components, including the molecular-beam valve, the sample, and the fiber-coupled desorption laser, which is movable in five axes, as required for experiments at central facilities. Utilizing strong-field ionization, we characterize the produced molecular beam and evaluate the influence of desorption laser pulse energy, relative timing of valve opening and desorption laser, sample bar height, and which part of the molecular packet is probed on the sample properties. Strong-field ionization acts as a universal probe and allows detecting all species present in the molecular beam, and hence enables us to analyze the purity of the produced molecular beam, including molecular fragments. We present optimized experimental parameters for the production of the purest molecular beam, containing the highest yield of intact parent ions, which we find to be very sensitive to the placement of the desorbed-molecule plumes within the supersonic expansion.