Weakly basic ions and polarizable bases favor SN2 over E2. Why? The argument used that polarizable nucleophiles form bonds earlier, stabilizing the transition state, can be applied to either SN2 or E2.
Polarisable Nucleophiles increase the rate of an Sn2-type reaction but not an E2 reaction because the more polarisable a base is, lower the electronegativity and charge density (conversely, higher ionic radius) it has.
Now, we know that the basic strength depends on these factors( in a period, we check the electronegativity and in a group, the ionic radius).
As, the Rate Determining Step of an E2 reaction involves the attack of a base, lower basic strength would mean that the reaction would be slower. Hence, more polatisable bases means that E2 would be slower.