In this paper, we present numerical and experimental results on Localized Surface Plasmon Resonance (LSPR) refractive index (RI) sensitivity, Figure of Merit (FoM), and penetration depth (dp) dependence on spherical gold nanoparticles (AuNPs) size, and the effects of AuNP dimer interparticle distance (ds) studied numerically. These parameters were calculated and observed for d = 20, 40, 60, 80, and 100 nm diameter spherical AuNPs. Our investigation shows d = 60 nm AuNPs give the best FoM. The AuNP dimer interparticle distance can significantly influence the RI sensitivity. Therefore, the effect of distances between pairs of d = 20 nm and 60 nm AuNPs is shown. We discuss the importance of penetration depth information for AuNPs functionalized with aptamers for biosensing in the context of aptamer size.