To balance, we write it in its half reactions and balance the half reactions by adding H2O, OH-, H+ and electrons. We do as follows:
MnO4-(aq) + Cl-(aq) --------> MnO2(s) + Cl2(g)
2 (2H2O + MnO2 --> MnO4- + 4H+ 3e- )
3 (2e- + Cl2 --> 2Cl- )
4H2O + 2MnO2 -----> 2MnO4- + 8H+ + 6e-
6e- + 3Cl2 -------> 6Cl-
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4H2O + 2MnO2 + 3Cl2 --------> 2MnO4- + 6Cl- + 8H+
Since it is in a basic solution,
8OH- + 2MnO2 + 3Cl2 -----> 2MnO4- + 6Cl- + 4H2O