From the NNDC, here are decay schemes for the two isotopes. Click to embiggen. There are a couple of important differences.

First, there's more energy to be released in the decay of $^{134}$Cs, $Q=\rm2.1\,MeV$, than in the decay of $^{137}$Cs, $Q=\rm1.2\,MeV$.
As a rule of thumb, more energetic decays proceed more rapidly.
However, the fact that neither decay goes directly to the ground state (the most energetic route) suggests there is a more important issue.
Second, the angular momentum considerations are more favorable in the $A=134$ system.
You can see that 97% of the decays of $^{134}$Cs, with ground state spin and parity $J^P=4^+$, are to the two $4^+$ levels in the daughter nucleus. These are so-called superallowed decays, where the angular momentum of the nucleus does not change and so zero net orbital or spin angular momentum needs to be carried away by the electron-neutrino pair.
By contrast, most of the $^{137}$Cs decays, from its $J^p=7/2^+$ ground state, are to the $11/2^-$ excited state in its daughter nucleus. This is a so-called first-forbidden decay, because the nuclear spin changes by $2\hbar$ and the parity changes as well. As the name suggests, these decays are slower than allowed or superallowed decays. A hand-waving way to think of it is that, since the decay products must have nonzero orbital angular momentum, their angular wavefunctions must be one of the $L>0$ spherical harmonics; however, those wavefunctions have much less overlap with the nucleus than the $L=0$, $s$-wave spherical harmonic.
You can see the shape of this if you look at some of the other decay channels.
The $^{134}$Cs decay to the $3^+$ state in the daughter occurs in 2.5% of decays; this is an "allowed" decay in the nomenclature, and its branching fraction corresponds to a "partial lifetime" of $\rm 2\,y/0.025 = 80\,y$, not grossly different from the first-forbidden lifetime in $^{137}$Cs.
There is also the electron-capture decay $\rm^{134}Cs\to{}^{134}Xe$, which has a lower $Q$-value, is "second-forbidden" due to angular momentum considerations, and has a branching fraction of $3\times10^{-4}$.
There are useful and correct comments in your other answers. It's true, as a rule, that odd-odd nuclei tend to be less stable than even-even nuclei --- in fact there are only four odd-odd nuclei that are stable, and only five more that are long-lived enough to occur naturally.
And it's reasonable to generalize this discussion: the decay rate tends to get faster if there are more available states to decay to, tends to get faster if there's more energy to release, and tends to get slower if more angular momentum change is required.