Balance atoms: (for everything except oxygen and hydrogen):
MnO4- → MnO2 (atoms are balanced)
Balance oxygen:
MnO4- → MnO2 + 2H2O
Balance hydrogen:
4H+ + MnO4- → MnO2 + 2H2O
Make adjustments for basic solution.
Add hydroxide ions to balance hydrogen:
4OH- + 4H+ + MnO4- → MnO2 + 2H2O + 4OH-
Remove hydrogen by forming water molecules and removing them:
4H2O + MnO4- → MnO2 + 2H2O + 4OH-
Cancel out any water molecules that are on both sides of the equation:
2H2O + MnO4- → MnO2 + 4OH-
Balance charge:
2H2O + MnO4- +3e- → MnO2 + 4OH-