Unstable surfaces (such as swiss balls, BOSU balls, wobble boards, or gymnastic rings) can be used to increase instability with strength training. There are several other ways to increase instability such as lifting with a single limb (arm or leg), off-set loading (like a kettlebell press), and increasing the distance the resistance is from the point of contact with the ground (or bench).
Instability is a 'tool' that you can add to exercises in your training program to help you accomplish your goals. Unstable surface training is neither good or bad itself. It just needs to be used properly. Unfortunately, most people don't use it properly. Think of a stable-unstable continuum. The more stability you have, the greater you can load a target muscle group and build muscle, strength, and power. On the other end, the more instability you have, the less you can load the target muscle group and the more you will recruit joint stabilizing muscles (especially at the ankle, hip, torso, and shoulder joints). An optimum level of stability is needed to gain the most out of your training and is based on your needs. A good trainer is like a craftsman, adeptly able to apply the right amount of instability to the appropriate joints at the appropriate times.
Unstable surface training is not the panacea of core training. In fact, it could limit your ability to progressively overload the torso muscles. Simply increasing resistance of most free weight exercises helps to strengthens the torso muscles. Let's look at four examples of the stability-instability continuum.
1. Dumbell shoulder press on an upright bench. High stability, greatest loading of arms and shoulders, and least loading of the stabilizers.
2. Standing Dumbell Shoulder Press. Less stability than the seated version. More demands of the torso and shoulder stabilizers.
3. Dumbell single-arm shoulder press. Even greater stabilzing demands placed on torso. Lesser loading of the arms and shoulders.
4. Dumbell single-arm shoulder press on a BOSU ball. The least loading of the arms and shoulders. Even more stabilizing demands of shoulders, torso, hips, and ankles:
These four different exercises demonstrate a shift in muscular loading and recruitment without changing the weight. You can increase loading of the target muscles AND stabilizers by simply increasing the amount of weight lifted. Adding an unstable surface will allow you to recruit more stabilizer muscles without having to add weight (which may be beneficial if you have an injury you are working around or rehabing).
The bottom line is that the body adapts to the specific musclular demands places upon it. Higher-trained athletes and lifters may see a negative effect by adding an unstable surface. Again, you need know the limitations of unstable surface training and apply it appropriately just like with other strength training 'tools'.