We present the results of high-precision UBV(RI)_c_ photometric observations and of spectroscopic radial velocity measurements obtained at the European Southern Observatory for a sample of 51 cool stars detected in the EUV by the ROSAT Wide Field Camera (WFC). Using also recent results from HIPPARCOS, we infer spectral types and investigate the single or binary nature of the sample stars. Optical variability, with periods in the 0.4-13 day range, has been detected for the first time in 15 of these stars.