  We study a quasi-Floquet state of a $\delta$-kicked rotor with absorbing
boundaries focusing on the nature of the dynamical localization in open quantum
systems. The localization lengths $\xi$ of lossy quasi-Floquet states located
near the absorbing boundaries decrease as they approach the boundary while the
corresponding decay rates $\Gamma$ are dramatically enhanced. We find the
relation $\xi \sim \Gamma^{-1/2}$ and explain it based upon the finite time
diffusion, which can also be applied to a random unitary operator model. We
conjecture that this idea is valid for the system exhibiting both the diffusion
in classical dynamics and the exponential localization in quantum mechanics.
