We study contact effects on electron transport across a molecular wire sandwiched between two semi-infinite (carbon) nanotube leads as a model for nanoelectrodes. Employing the Landauer scattering matrix approach, we find that the conductance is very sensitive to parameters such as the coupling strength and geometry of the contact. The conductance exhibits markedly different behavior in the two limiting scenarios of single contact and multiple contacts between the molecular wire and the nanotube interfacial atoms. In contrast to a single contact, the multiple-contact configuration acts as a filter selecting single transport channels. It exhibits a scaling law for the conductance as a function of coupling strength and tube diameter. We also observe an unusual narrow-to-broad-to-narrow behavior of conductance resonances upon decreasing the coupling.