Identify the oxidation and reduction half-reactions. In this case, ClO⁻ is reduced to Cl⁻, and Cr(OH)₄⁻ is oxidized to CrO₄²⁻.
Balance the atoms other than O and H in each half-reaction. For the reduction half-reaction, balance Cl atoms. For the oxidation half-reaction, balance Cr atoms.
Balance the oxygen atoms by adding H₂O molecules to the side that needs oxygen. For the reduction half-reaction, add H₂O to the side with Cl⁻. For the oxidation half-reaction, add H₂O to the side with CrO₄²⁻.
Balance the hydrogen atoms by adding OH⁻ ions to the side that needs hydrogen. Since the solution is basic, add OH⁻ to both sides to neutralize the H⁺ ions formed.
Combine the balanced half-reactions, ensuring that the electrons cancel out. Adjust coefficients to ensure that the number of electrons lost in oxidation equals the number gained in reduction.