Depends on what sort of car you're driving.
Something with huge torque (like the speed 12 or Zonda, for example) can get away without too much downshifting, because they'll accelerate just as well in their torque band as they will in their power band.
For a car that has high power but low torque, you'll basically have to "row" the gearshift (i.e. downshift often) to keep the engine in its power zone.
Another thing to consider when downshifting as well is the engine braking effect. As you go down gears, you are decreasing your engine to wheel gearing. This multiplies the power going out of the gearbox when accelerating, but it also increases the amount of force the wheels will have to generate to 'drive' the engine (back compression). So, the lower the gear you are in, the stronger your engine braking will be. That means you don't have to force the brakes as hard because your engine is taking some of the braking load.
So, for braking and cornering, I definitely recommend downshifting, and often. Get to as low a gear as practical and use the engine braking/acceleration. However, if you are in the torque band and the car is accelerating fine, then there's no need to change down gears.