1) Split the equation into half-reactions:
ClO3(-) --> Cl(-)
AsO2(-) --> AsO4(3-)
2) Do the following to both half-reactions:
a) Balance non O and H atoms first (in this case, they are already balanced)
b) Balance O atoms by adding H2O to the side that needs more O atoms
ClO3(-) --> Cl(-) + 3H2O
AsO2(-) + 2H2O --> AsO4(3-)
c) Balance H atoms by adding H+ to the side needing more hydrogens
ClO3(-) + 6H+ --> Cl(-) + 3H2O
AsO2(-) + 2H2O --> AsO4(3-) + 4H+
d) Balance the charge so that the charge on one side of the half-reaction equals the charge on the other side (charge of reactants=charge of products for each half-reaction). Do this by adding electrons to the more positively charged side:
ClO3(-) + 6H+ + 6e(-) --> Cl(-) + 3H2O <-- electrons on the reactant side indicates this half-reaction is being reduced.
AsO2(-) + 2H2O --> AsO4(3-) + 4H+ + 2e(-) <-- electrons on the product side indicates this half-reaction is being oxidized.
We can use this method to determine which half-reaction is being reduced and which is being oxidized. Although, you can usually tell just by looking at the half-reactions.
3) Since the amount of electrons released from oxidation must equal the number of electrons accepted by the reduced species, we need to multiply the half-reactions so the electrons added in each half-reaction equal each other:
1(ClO3(-) + 6H+ + 6e(-) --> Cl(-) + 3H2O)
3(AsO2(-) + 2H2O --> AsO4(3-) + 4H+ + 2e(-))
4) Combine the half-reactions back into one equation and simplify the equation:
ClO3(-) + 6H+ + 6e(-) + 3AsO2 + 6H2O --> Cl- + 3H2O + 3AsO4(3-) + 12H+ + 6e(-)
After canceling, we have the answer:
ClO3(-) + 3AsO2 + 3H2O --> Cl(-) + 3AsO4(3-) + 6H+
*NOTE: If this solution is acidic, then we stop here and the equation above is our answer, but if done in basic solution, we need to do a few more steps. Add OH- to the side with H+ to render H20 molecules. Then re-simplify the equation and you're done. You add OH- because the solution is basic, and H+ won't usually exist in a basic solution.
Hope this helps. Sorry if it is long or confusing.