Quasi-stellar objects (QSOs) occur in galaxies in which supermassive black holes (SMBHs) are growing substantially through rapid accretion of gas. Many popular models of the co-evolutionary growth of galaxies and black holes predict that QSOs are also sites of substantial recent star formation (SF), mediated by important processes, such as major mergers, which rapidly transform the nature of galaxies. A detailed study of the star-forming properties of QSOs is a critical test of these models. We present a far-infrared Herschel/PACS study of the mean star formation rate (SFR) of a sample of spectroscopically observed QSOs to z ~ 2 from the COSMOS extragalactic survey. This is the largest sample to date of moderately luminous QSOs (with nuclear luminosities that lie around the knee of the luminosity function) studied using uniform, deep far-infrared photometry. We study trends of the mean SFR with redshift, black hole mass, nuclear bolometric luminosity, and specific accretion rate (Eddington ratio). To minimize systematics, we have undertaken a uniform determination of SMBH properties, as well as an analysis of important selection effects of spectroscopic QSO samples that influence the interpretation of SFR trends. We find that the mean SFRs of these QSOs are consistent with those of normal massive star-forming galaxies with a fixed scaling between SMBH and galaxy mass at all redshifts. No strong enhancement in SFR is found even among the most rapidly accreting systems, at odds with several co-evolutionary models. Finally, we consider the qualitative effects on mean SFR trends from different assumptions about the SF properties of QSO hosts and from redshift evolution of the SMBH-galaxy relationship. While currently limited by uncertainties, valuable constraints on AGN-galaxy co-evolution can emerge from our approach.