Consider, for instance, a simple system that has only two energy levels \epsilon_0 = 0 and \epsilon_1 = \Delta E .
For a system governed by the Boltzmann distribution we find (the partition function has been taken out)
W(0\rightarrow 1)\exp{-(\epsilon_0/kT)} = W(1\rightarrow 0)\exp{-(\epsilon_1/kT)}.We get then
\frac{W(1\rightarrow 0)}{W(0 \rightarrow 1)}=\exp{-(\Delta E/kT)},which goes to zero when T tends to zero.