The Kolmogorov distribution is defined by the distribution of the random variable $K:=\sup_{0\leqslant t\leqslant 1}|B(t)|$, where $B(t)$ is the Brownian Bridge. 

The problem of existence of moments for $K$ is actually the same as the study of moments of $K':=\sup_{0\leqslant t\leqslant 1}|W(t)|$, where $W(t)$ is a standard Brownian motion. An application of [Doob's (sub)martingale inequality][1] gives that for all $C>0$,
$$P(K'\geqslant C)\leqslant \exp\left(-\frac{C^2}2\right).$$

Using the fact that for a non-negative random variable $X$ and $p>1$, we have $$E(X^p)=\int_0^{+\infty}pt^{p-1}P(X\geqslant t)dt,$$
we conclude that Kolmogorov distribution admits moments of any order.

  [1]: https://en.wikipedia.org/wiki/Doob's_martingale_inequality