Kolmogorov backwards equation / stationary distribution

One can in the case of the Fokker-Planck / forward Kolmogorov equation, set the time derivative term to zero, and solve the remaining ODE to obtain the "forward-time" stationary distribution.

Does it make any sense to solve for the stationary distribution of the backward Kolmogorov equation? If so, what would be the interpretation?

The equation $Lu=0$ where $L$ is the generator of your process (the formal adjoint of the Fokker-Planck operator) does indeed have significance, but its significance depends on the boundary conditions that you impose. In the case of the stationary FPE, the natural "boundary condition" is just the normalization and positivity conditions. About the only other thing that can make sense is dropping the normalization condition (so that you are finding a stationary measure which is not a distribution).
A special case of this is when you have two open sets $A,B$ with disjoint closures which are in the domain of the process but are excluded from the domain of your BKE. You impose the Dirichlet constraint of zero on the boundary of $A$ and of one on the boundary of $B$. Then the solution is called the committor, and it measures the probability of hitting $B$ before hitting $A$ starting from a given point $x$. It can be used to compute the rate of the process of going from $A$ to $B$ without returning to $A$, which is a kind of "reaction rate". This is used in "transition path theory".