J.R. S. answered 10/23/19
Ph.D. University Professor with 10+ years Tutoring Experience
NOTE: The following is based on the equation as you have posted it, but it may be that you meant the have I- as a product and not IO-. If that's the case, the answer will obviously be different, and you should re-post or make a comment to the current question/answer for follow up. Also, note in your answer that the oxygens, hydrogens and charge are not balanced, so it couldn't possibly be correct. You should always check that mass and charge are the same on both sides of the equation.
IO3- ===> IO-
IO3- ===> IO- + 2H2O ... to balance oxygens
IO3- + 4H+ ===> IO- + 2H2O ... to balance hydrogens
IO3- + 4H+ + 4OH- ===> IO- + 2H2O + 4OH- ... because it is in basic solution
IO3- + 4H+ + 4OH- + 4e- ===> IO- + 2H2O + 4OH- ... to balance charge
IO3- + 2H2O + 4e- ===> IO- + 4OH- ... REDUCTION HALF REACTION
Re ===> ReO4
Re + 4H2O ===> ReO4 ... to balance oxygens
Re + 4H2O ===> ReO4 + 8H+ ... to balance hydrogens
Re + 4H2O + 8 OH- ===> ReO4 + 8H+ + 8OH- ... because it is in basic solution
Re + 4H2O + 8 OH- ===> ReO4 + 8H+ + 8OH- + 8e- ... to balance charge
Re + 8OH- ===> ReO4 + 4H2O + 8e- ... OXIDATION HALF REACTION
Multiply reduction half reaction by 2 to equalize the electrons and add the half reactions to get:
2IO3- + 4H2O + 8e- + Re + 8OH- ===> 2IO- + 8OH- + ReO4 + 4H2O + 8e-
Finally, cancel common items on each side and condense to get ...
2IO3- + Re ==> 2IO- + ReO4 which is balanced for both mass and charge