We have investigated optical conductivity spectra of $n$- and $p$-type Ba$_8$Ga $_{16}$Sn$_{30}$ ($alpha$-BGS) with type-VIII clathrate structure, at temperatures from 296,K down to 6,K with a terahertz time-domain spectrometer (0.2,-,2.5,THz). The continuous spectra contributed from charge carriers are dispersive in this frequency range and also temperature- and carrier type-dependent. The Drude-Smith model taking multiple-scatterings of charge carriers into account well reproduces those data. The relaxation rate of the $n$-type carriers decreases more sharply than that in the $p$-type material, suggesting that a stronger electron-phonon interaction may exist in the $n$-type than in the $p$-type. On the other hand, the localized infrared-active modes observed at 1.3,THz and 1.7,THz, identified as the rattling phonons of the Ba$^{2+}$ ions quasi-on-center vibrations, become soft and broad significantly with decreasing temperature as well as observed in type-I BGS and BGG (Ba$_8$Ga$_{16}$Ge$_{30}$) clathrates. The softening in the $n$-type is smaller by about 30% than in the $p$-type, whereas the linewidth brodening is almost the same independently on the carrier type. The difference in the softening is discussed with a scenario where the interaction of rattling phonons with carriers can modify the anharmonic potential of the guest ions. The anomalous broadening at low temepratures is also discussed by the impurity-scattering model presented for a rattling-phonon system strongly hybridized with acoustic cage phonons.
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