A) Separate your two half reactions:
ClO3-1 --> Cl-1 ...Cl's oxidation # changes from +5 to -1 = gain of 6e-
AsO2-1 --> AsO4 ...As's oxidation # changes from +3 to +5 = loss of 2e-
B) Start with balancing the least abundant elements (Cl and As):
ClO3-1 --> Cl-1
AsO2-1 --> AsO4 3-
C) Proceed with balancing Oxygen by adding H2O:
ClO3-1 --> Cl-1 + 3H2O
AsO2-1 + 2H2O --> AsO4 3-
D) Balance H by adding H+:
ClO3-1 + 6H+ --> Cl-1 + 3H2O
AsO2-1 + 2H2O --> AsO4 3- + 4H+
E) Balance the charge by adding electrons:
ClO3-1 + 6H+ + 6e- --> Cl-1 + 3H2O
AsO2-1 + 2H2O --> AsO4 3- + 4H+ + 2e-
F) Multiply by proper coefficients to cancel out e-:
1[ClO3-1 + 6H+ + 6e- --> Cl-1 + 3H2O]
3[AsO2-1 + 2H2O --> AsO4 3- + 4H+ + 2e-]
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ClO3-1 + 3AsO2-1 + 3H2O --> Cl-1 + 6H+ + 3AsO4 3-