Context.A sample of 14 young open star clusters has been observed in the TeV energy regime with the stereoscopic system of the HEGRA (High Energy Gamma-Ray Astronomy) Cherenkov telescopes from 1997 to 2002, resulting in more than 300 h of observation time. Aims.Young open star clusters may contribute to the acceleration of cosmic rays. The detection of γ-rays (from decaying $\pi^0$s produced in hadronic interactions) from these objects could be evidence for such a contribution. The results of our observations are compared to available γ-ray data and to a simple hadronic model in the framework of shock front acceleration of cosmic rays in the stellar winds of the cluster members to test the potential of the presently available data on young open star clusters to constrain this type of model. Methods.The stereoscopic system of HEGRA Cherenkov telescopes makes use of the atmospheric imaging technique. Air showers initiated by primary Gamma-Rays are recorded as elliptical images in the telescope cameras. The images from the different telescopes are then superimposed to reconstruct the parameters of the primary particle. This technique (stereoscopy) was pioneered by the HEGRA experiment. Results.No significant excess has been found in the analysed data set of young open star clusters. The derived upper limit on the TeV gamma-ray flux from Berkeley 87 and the available EGRET data from the same direction do not allow us to fully constrain the simple hadronic model used here. The comparison of the upper limits derived for all 14 objects with the flux detected from TeV J2032+4130 (under the assumption of an association of the TeV-signal with the compact stellar association Cyg OB2) suggests that γ-ray emission from young open star clusters as an object class cannot be ruled out.