We describe the combination of buffer-gas cooling with electrostatic velocity filtering to produce a high-flux continuous guided beam of internally cold and slow polar molecules. In a previous paper (L.D. van Buuren et al., Phys. Rev. Lett., 2009, 102, 033001) we presented results on density and state purity for guided beams of ammonia and formaldehyde using an optimized set-up. Here we describe in more detail the technical aspects of the cryogenic source, its operation, and the optimization experiments that we performed to obtain the best performance. The versatility of the source is demonstrated by the production of guided beams of different molecular species.