We study the anomalous $ZZH$ vertex contributing to the Higgsstrahlung process at the LHC and future $e^+e^-$ colliders using the polarization of the $Z$ as a probe. We calculate the eight independent polarization parameters of the $Z$ boson by using the spin density matrix formalism. We estimate 1$\sigma$ limits on the anomalous $ZZH$ couplings using the angular asymmetries of leptons from $Z$ decay which are simply related to the eight polarization parameters. We discuss the results for a) two possible c.m.energies $(\sqrt{s}=)$ 250 GeV and 500 GeV for the $e^+e^-$ colliders, with the possibility of polarized $e^+e^-$ beams, b) LHC at $\sqrt{s}=14\text{ TeV}$.