We present a highly complete sample of broad-line (Type 1) QSOs out to z~3 selected by their mid-infrared colors, a method that is minimally affected by dust reddening. We remove host-galaxy emission from the spectra and fit for excess reddening in the residual QSOs, resulting in a Gaussian distribution of colors for unreddened (blue) QSOs, with a tail extending toward heavily reddened (red) QSOs, defined as having E(B-V)>0.25. This radio-independent selection method enables us to compare red and blue QSO radio properties in both the FIRST (1.4GHz) and VLASS (2-4GHz) surveys. Consistent with recent results from optically selected QSOs from SDSS, we find that red QSOs have a significantly higher detection fraction and a higher fraction of compact radio morphologies at both frequencies. We employ radio stacking to investigate the median radio properties of the QSOs including those that are undetected in FIRST and VLASS, finding that red QSOs have significantly brighter radio emission and steeper radio spectral slopes compared with blue QSOs. Finally, we find that the incidence of red QSOs is strongly luminosity dependent, where red QSOs make up >40% of all QSOs at the highest luminosities. Overall, red QSOs comprise ~40% of higher luminosity QSOs, dropping to only a few percent at lower luminosities. Furthermore, red QSOs make up a larger percentage of the radio-detected QSO population. We argue that dusty AGN-driven winds are responsible for both the obscuration as well as excess radio emission seen in red QSOs.