We present a temperature- and polarization-resolved phononic and electronic Raman scattering study in combination with the first-principles calculations on the kagome metal Ni3In with anisotropic transport properties and non-Fermi liquid behavior. At temperatures below 50 K and down to 2 K, several Raman phonon modes, including particularly an interlayer shear mode, exhibit appreciable frequency and linewidth renormalization, reminiscent of the onset of the Kondo screening without an accompanying structural or magnetic phase transition. In addition, a low-energy electronic continuum observed in polarization perpendicular to the kagome planes reveals strong temperature dependence below 50 K, implying thermal depletion of incoherent quasiparticles, while the in-plane continuum remains invariant. These concomitant electronic and phononic Raman signatures suggest that Ni3In undergoes an anisotropic electronic crossover from an incoherent to a coherent Kondo lattice regime below 50 K. We discuss the origin of the anisotropic incoherent-coherent crossover in association with the possible anisotropic Kondo hybridization involving localized Ni-3dxz flat-band electrons.