We present Chandra X-ray observations of 14 radio-loud quasars at redshifts $3 < z < 4$, selected from a well-defined sample. All quasars are detected in the 0.5-7.0 keV energy band, and resolved X-ray features are detected in five of the objects at distances of 1-12 from the quasar core. The X-ray features are spatially coincident with known radio features for four of the five quasars. This indicates that these systems contain X-ray jets. X-ray fluxes and luminosities are measured, and jet-to-core X-ray flux ratios are estimated. The flux ratios are consistent with those observed for nearby jet systems, suggesting that the observed X-ray emission mechanism is independent of redshift. For quasars with undetected jets, an upper limit on the average X-ray jet intensity is estimated using a stacked image analysis. Emission spectra of the quasar cores are extracted and modeled to obtain best-fit photon indices, and an Fe K emission line is detected from one quasar in our sample. We compare X-ray spectral properties with optical and radio emission in the context of both our sample and other quasar surveys.