Context. Several observations of stellar jets show evidence of X-ray emitting shocks close to the launching site. In some cases, including young stellar objects (YSOs) at different stages of evolution, the shocked features appear to be stationary. We study two cases, both located in the Taurus star-forming region. HH 154, the jet originating from the embedded binary Class 0/I protostar IRS 5, and the jet associated with DG Tau, a more evolved Class II disk-bearing source or classical T Tauri star (CTTS). Aims. We investigate the effect of perturbations in X-ray emitting stationary shocks in stellar jets and the stability and detectability in X-rays of these shocks, and we explore the differences in jets from Class 0 to Class II sources. Methods. We performed a set of 2.5D magnetohydrodynamic numerical simulations that model supersonic jets ramming into a magnetized medium. The jet is formed of two components: a continuously driven component that forms a quasi-stationary shock at the base of the jet and a pulsed component consisting of blobs perturbing the shock. We explored different parameters for the two components. We studied two cases: HH 154, a light jet (less dense than the ambient medium), and a heavy jet (denser than the ambient medium) associated with DG Tau. We synthesized the count rate from the simulations and compared these data with available Chandra observations. Results. Our model is able to reproduce the observed jet properties at different evolutionary phases (in particular, for HH 154 and DG Tau) and can explain the formation of X-ray emitting quasi-stationary shocks observed at the base of jets in a natural way. The jet is collimated by the magnetic field forming a quasi-stationary shock at the base which emits in X-rays even when perturbations formed by a train of blobs are present. We found similar collimation mechanisms dominating in both heavy and light jets. Conclusions. We derived the physical parameters that can give rise to X-ray emission consistent with observations of HH 154 and DG Tau. We have also performed a wide exploration of the parameter space characterizing the model; this can be a useful tool to study and diagnose the physical properties of YSO jets over a broad range of physical conditions, from embedded to disk-bearing sources. We show that luminosity does not change significantly in variable jet models for the range of parameters explored. Finally, we provide an estimation of the maximum perturbations that can be present in HH 154 and DG Tau taking into account the available X-ray observations.