Why does KMnO4 oxidize iodide to iodine in acidic medium, whereas iodide to iodate in alkaline medium?
A simple way to explain this is to illustrate with the following Latimer diagrams:



If you calculate the standard electrode potential of the highlighted portions it will be positive, so the reaction is spontaneous under standard conditions.

For acidic conditions, iodine is not converted to iodate because the electrode potential of the reaction would become negative, whereas for basic conditions it is possible.

Note that both the manganate and the iodide ions are both affected by acidic and basic conditions; thus it is important to consider their respective electrode potentials.

(Data was taken from Inorganic Chemistry, Shriver et al., Sixth Edition, Oxford University Press)