We study the modification of the classical criterion for the linear onset and growing rate of the Rayleigh-Taylor instability (RTI) in a partially ionized plasma in the two-fluid description. The plasma is composed of a neutral fluid and an electron-ion fluid, coupled by means of particle collisions. The governing linear equations and appropriate boundary conditions, including gravitational terms, are derived and applied to the case of the RTI in a single interface between two partially ionized plasmas. The limits of collisionless, no gravity, and incompressible fluids are checked before addressing the general case. We find that both compressibility and ion-neutral collisions lower the linear growth rate, but do not affect the critical threshold of the onset of the RTI. The configuration is always unstable when a lighter plasma is below a heavier plasma regardless the value of the magnetic field strength, the ionization degree, and the ion-neutral collision frequency. However, ion-neutral collisions have a strong impact on the RTI growth rate, which can be decreased by an order of magnitude compared to the value in the collisionless case. Ion-neutral collisions are necessary to accurately describe the evolution of the RTI in partially ionized plasmas such as prominences. The timescale for the development of the instability is much longer than in the classical incompressible fully ionized case. This result may explain the existence of prominence fine structures with life times of the order of 30 minutes. The timescales derived from the classical theory are about one order of magnitude shorter and incompatible with the observed life times.