Aims. We study the H₂O chemistry in star-forming environments under the influence of a central X-ray source and a central far ultraviolet (FUV) radiation field. The X-ray models are applied to envelopes around low-mass Class 0 and I young stellar objects (YSOs). Methods. The gas-phase water chemistry is modeled as a function of time, hydrogen density and X-ray flux. To cover a wide range of physical environments, densities between n_H = 10⁴-10⁹ cm^-3 and temperatures between T = 10-1000 K are studied. Results. Three different regimes are found: for T < 100 K, the water abundance is of order 10^-7-10^-6 and can be somewhat enhanced or reduced due to X-rays, depending on time and density. For 100 K ≲ T ≲ 250 K, H₂O is reduced from initial x(H ₂O) ≈10^-4 following ice evaporation to x(H ₂O) ≈ 10^-6 for F_X ≳ 10^-3 erg s^-1 cm^-2 (t = 10⁴ yr) and for F_X ≳ 10^-4 erg s^-1 cm^-2 (t = 10⁵ yr). At higher temperatures (T ≳ 250 K) and hydrogen densities, water can persist with x(H₂O) ≈ 10^-4 even for high X-ray fluxes. Water is destroyed in both Class 0 and I envelopes on relatively short timescales (t ≈ 5000 yr) for realistic X-ray fluxes, although the effect is less prominent in Class 0 envelopes due to the higher X-ray absorbing densities there. FUV photons from the central source are not effective in destroying water. Conclusions. X-rays reduce the water abundances especially in regions where the gas temperature is T ≲ 250-300 K for fluxes F_X ≳ 10^-5-10^-4 erg s^-1 cm^-2. The affected regions can be envelopes, disks or outflow hot spots. The average water abundance in Class I sources for L_X ≳ 10²⁷ erg s ^-1 is predicted to be x(H₂O) ≲ 10^-6. Central UV fields have a negligible influence, unless the photons can escape through cavities.