Electron collisions for a two dimensional Fermi liquid (FL) are shown to give a quasiparticle damping with interesting frequency and temperature variations in the BCS superconducting state. The spin susceptibility which determines the structure of the damping is analyzed in the normal state for a Hubbard model with a constant on--site Coulomb repulsion. This is then generalized to the superconducting state by including coherence factors and self energy and vertex corrections. Calculations of the NMR relaxation rate reveal that the FL damping structure can reduce the Hebel--Slichter peak, in agreement with data on the organic superconductor (MDT-TTF)$_2$AuI$_2$. However, the strongly suppressed FL damping in the superconducting state does not eliminate the Hebel-Slichter peak, and thus suggests that other mechanisms are needed to explain the NMR data on (TMTSF)$_2$ClO$_4$, the BEDT organic compounds, and cuprate superconductors. Predictions of the temperature variation of the damping and the spin response are given over a wide frequency range as a guide to experimental probes of the symmetry of the superconducting pairs.

10 pages, RevTeX 3.0, 9 figures in uuencoded postscript