Almost any rubber product will degrade over time, loosing its properties. Different rubber compounds will degrade differently, depending on the formula and method of fabrication and, of course, environmental factors. This applies to brake pads, tires, grips, valve seals, inner tubes and more.
Usually brake pads tend to "dry" over time, turning less flexible and prone to cracks. A less flexible pad is less capable of creating friction. I know of some rubber (or rubber-like) products that when degraded turn to a thick paste that will stain fabrics, skin, etc. but never seen a brake pad degrade this way, if that where the case, it would not last eve one braking.
To test a V-Brake pad press hard with your fingernail on its working surface. You should be able to sink the fingernail just a little bit, leaving a pressure mark that will shortly disappear. If it feels hard, wood-like, almost sure you are better discarding it.
However, pad age is not the only factor that can degrade braking performance. Dirt and grease on pad or rim for example. I recommend washing them with liquid dish washing soap. Another factor is calibration. Brake pads not properly aligned may will not perform as designed.
Finally, there are different rubber formulas for aluminum or steel rims, (And for carbon fiber I guess) and I have tested pads that work wonders on steel are worthless on aluminum. Also have observed that the same pad can perform differently on aluminum rims made of different alloys.
<< End of answer and begin of anecdote >>
Recently I committed the mistake of trying to slowly descend a long pronounciated slope with a road bike from 1984 that still had the original pads (The ones it was originally sold with). The result was the friction generated was so poor that they only caused the rims to heat up to the point that both inner tubes were blown out. One rim was steel and the other a single wall aluminum.