How does one determine whether to use an acidic or basic solution for balancing redox reactions?

In any redox reaction, you are attempting to balance out the acids and bases (i.e. using reducing and oxidizing agents). If you can calculate the moles of acids or bases that are on the left-hand side of your equation, and determine which is the "limiting factor" (i.e. which one will 'run out' first and not be left in excess), you can calculate from that the amount of the opposing solution (base to counter-balance acid and acid to counter-balance base) that you would need to add to balance the equation.

In general you must be told -- you can't really choose. All redox reactions can be balanced under either acidic or basic conditions.

As a practical matter some reactions only occur under one of those conditions, but you need to understand the particular reaction in detail to know which.