The $\jpsi$ $\pt$ spectrum and nuclear modification factor ($\raa$) are reported for $\pt < 5 \ \gevc$ and $|y|<1$ from 0\% to 60\% central Au+Au and Cu+Cu collisions at $\snn = 200 \ \gev$ at STAR. A significant suppression of $\pt$-integrated $\jpsi$ production is observed in central Au+Au events. The Cu+Cu data are consistent with no suppression, although the precision is limited by the available statistics. $\raa$ in Au+Au collisions exhibits a strong suppression at low transverse momentum and gradually increases with $\pt$. The data are compared to high-$\pt$ STAR results and previously published BNL Relativistic Heavy Ion Collider results. Comparing with model calculations, it is found that the invariant yields at low $\pt$ are significantly above hydrodynamic flow predictions but are consistent with models that include color screening and regeneration.