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The stable compactification of the moduli space of parametrized $J$-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^\\infty$-topology. We construct a Kuranishi structure with corners in the sense of Fukaya and Ono. This Kuranishi structure is orientable if $L$ is spin. 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The stable compactification of the moduli space of parametrized $J$-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^\\infty$-topology. We construct a Kuranishi structure with corners in the sense of Fukaya and Ono. This Kuranishi structure is orientable if $L$ is spin. 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