By using Lanczos exact diagonalization and quantum Monte Carlo combined with stochastic analytic continuation, we study the dynamical properties of the $S=1$ antiferromagnetic Heisenberg chain with different strengths of bond disorder. In the weak disorder region, we find weakly coupled bonds which can induce additional low-energy excitation below the one-magnon mode. As the disorder increases, the average Haldane gap closes at $delta_{Delta}sim 0.5$ with more and more low-energy excitations coming out. After the critical disorder strength $delta_csim 1$, the system reaches a random-singlet phase with prominent sharp peak at $omega=0$ and broad continuum at $omega>0$ of the dynamic spin structure factor. In addition, we analyze the distribution of random spin domains and numerically find three kinds of domains hosting effective spin-1/2 quanta or spin-1 sites in between. These spins can form the weakly coupled long-range singlets due to quantum fluctuation which contribute to the sharp peak at $omega=0$.
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