Superconductivity-induced changes in the electronic Raman scattering response were observed for the RNi$_2$B$_2$C (R = Y, Lu) system in different scattering geometries. In the superconducting state, 2$\Delta$-like peaks were observed in A$_{1g}$, B$_{1g}$, and B$_{2g}$ spectra from single crystals. The peaks in A$_{1g}$ and B$_{2g}$ symmetries are significantly sharper and stronger than the peak in B$_{1g}$ symmetry. The temperature dependence of the frequencies of the 2$\Delta$-like peaks shows typical BCS-type behavior, but the apparent values of the $2\Delta$ gap are strongly anisotropic for both systems. In addition, for both YNi$_2$B$_2$C and LuNi$_2$B$_2$C systems, there exists reproducible scattering strength below the $2\Delta$ gap which is roughly linear to the frequency in B$_{1g}$ and B$_{2g}$ symmetries. This discovery of scattering below the gap in non-magnetic borocarbide superconductors, which are thought to be conventional BCS-type superconductors, is a challenge for current understanding of superconductivity in this system.