We report on magnetoresistance measurements of YBa2Cu3O7/PrBa2Cu3O7 (YBCO/PrBCO) superlattices in fields up to 12 T at temperatures below, but not too far from, the superconducting transition temperature T(c). By varying the thickness of the PrBCO separator layer, the influence of the interlayer coupling on the superconducting properties is studied. An excellent fitting by the Larkin two-dimensional superconducting fluctuation theory with only two scaling parameters (A(T) and H(PHI)(T)) for each R(T) curve shows that by taking thicker PrBCO layers it is possible to induce the dimensional crossover from the anisotropic 3D behavior in pure YBCO to the quasi-2D behavior in YBCO/PrBCO superlattices. The resistive transition broadening in a magnetic field can be related to giant conductivity fluctuations in the quasi-two-dimensional CuO2 superconducting double layers in YBCO. For Y:Pr = 1:3 and 1:5 superlattices, the temperature dependence of the first scaling parameter A(T), determined by the fluctuation amplitude, follows quite well the Larkin beta(T) function typical for 2D systems. For Y:Pr = 1:1 and 5:6 superlattices, the YBCO layers are not completely decoupled or too thick and A(T) deviates from beta(T). The temperature dependence of the second scaling parameter H(PHI)(T), the phase coherence breaking field, is consistent with the recent theoretical prediction by Reizer for the quasi-two-dimensional electron-electron interactions. The resistive transition tail, where the characteristic resistance is much smaller than the normal state resistance, is not directly caused by these fluctuations but rather is related to thermally assisted flux motion.