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In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_q\left(𝔮\left(n\right)\right)$. We define the notion of crystal bases and prove the tensor product rule for $U_q\left(𝔮\left(n\right)\right)$-modules in the category ${𝒪}_{\mathrm{int}}^{\ge 0}$. Our main theorem shows that every $U_q\left(𝔮\left(n\right)\right)$-module in the category ${𝒪}_{\mathrm{int}}^{\ge 0}$ has a unique crystal basis.