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We present high-precision ab initio calculations of the static and dynamic polarizability of the barium monohydroxide ($^{138}$BaOH) molecule, using relativistic coupled-cluster theory. By thoroughly investigating the dependence of the calculated polarizabilities on computational parameters (basis set size, treatment of relativity, level of treatment of electron correlation, and vibrational corrections), a procedure to determine uncertainties is constructed and applied. The dipole moment of BaOH is also calculated and compared to experiment, confirming the accuracy of the predicted polarizabilities. The static and dynamic ($λ=1064\;\text{nm}$) polarizability was calculated for both the ground state and the (010) vibrational bending mode, the latter state being particularly interesting for a wide range of quantum experiments. The ground state static polarizabilities were calculated to be 200.8(24) a.u. and 297(5) a.u. for the parallel and perpendicular components respectively.