Radio emission in the brightest radio quasars can be attributed to processes inherent to active galactic nuclei (AGN) powered by super massive black holes (SMBHs), while the physical origins of the radio fluxes in quasars without radio detections have not been established with full certainly. Deep radio surveys carried out with the Low Frequency ARray (LOFAR) are at least one order of magnitude more sensitive for objects with typical synchrotron spectra than previous wide-area high-frequency surveys ( > 1.0 GHz). With the enhanced sensitivity that LOFAR offers, we investigate the radio-infrared continuum of LOFAR radio-detected quasars (RDQs) and LOFAR radio-undetected quasars (RUQs) in the 9.3 deg2 NOAO Deep Wide-field survey (NDWFS) of the Boötes field; RUQs are quasars that are individually undetected at a level of ≥5σ in the LOFAR observations. To probe the nature of the radio and infrared emission, where direct detection is not possible due to the flux density limits, we used a median image stacking procedure. This was done in the radio frequencies of 150 MHz, 325 MHz, 1.4 GHz and 3.0 GHz, and in nine infrared bands between 8 and 500 μm. The stacking analysis allows us to probe the radio-luminosity for quasars that are up to one order of magnitude fainter than the ones detected directly. The radio and infrared photometry allow us to derive the median spectral energy distributions of RDQs and RUQs in four contiguous redshift bins between 0 < z < 6.15. The infrared photometry is used to derive the infrared star-formation rate (SFR) through SED fitting, and is compared with two independent radio-based star-formation (SF) tracers using the far-infrared radio correlation (FIRC) of star-forming galaxies. We find a good agreement between our radio and infrared SFR measurements and the predictions of the FIRC. Moreover, we use the FIRC predictions to establish the level of the contribution due to SMBH accretion to the total radio-luminosity. We show that SMBH accretion can account for ∼5−41% of the total radio-luminosity in median RUQs, while for median RDQs the contribution is ∼50−84%. This implies that vigorous SF activity is coeval with SMBH growth in our median stacked quasars. We find that median RDQs have higher SFRs that agree well with those of massive star-forming main sequence galaxies, while median RUQs present lower SFRs than RDQs. Furthermore, the behavior of the radio-loudness parameter (R = log10(Lrad/LAGN)) is investigated. For quasars with R ≥ −4.5, the radio-emission is consistent with being dominated by SMBH accretion, while for low radio luminosity quasars with R < −4.5 the relative contribution of SF to the radio fluxes increases as the SMBH component becomes weaker. We also find signatures of SF suppression due to negative AGN feedback in the brightest median RDQs at 150 MHz. Finally, taking advantage of our broad spectral coverage, we studied the radio spectra of median RDQs and RUQs. The spectral indices of RUQs and RDQs do not evolve significantly with redshift, but they become flatter towards lower frequencies.