Your question is a good one. Excessive stress can damage a joint (or nearby structures) and joint damage can lead to arthritis. For example, repeated sprains or cartilage injuries to the knee among football players may lead to arthritis.
Running certainly stresses weight-bearing joints. Your knees support up to seven times your weight while jogging. The stress is even higher with jumping or suddenly starting and stopping, like the motions used in basketball.
But, joints are not like tires. They do not wear out from use. The wear and tear type of arthritis (called osteoarthritis) is more likely from age, obesity, injury and genetics.
Running may not stress the joints enough to cause arthritis. There is no evidence that common, repetitive movements among recreational cyclists or joggers will damage or wear out the joints. This goes for most other repetitive movements, such as walking, painting or knitting. Repetitive motion is more likely to cause tendonitis than arthritis.
Research on repetitive physical activity and arthritis is difficult to perform. People who are active often differ from those who are sedentary in important ways. Studies that compare the two groups may come to faulty conclusions. While the results of the available studies are somewhat mixed, long-term runners are not more likely to wear out their weight-bearing joints than people who are sedentary.
A large study of runners was published in 1998. It found that over a nine-year period, members of a running club had no higher incidence of osteoarthritis than a similar group of non-runners. A more recent study found that long-distance runners developed less osteoarthritis over 20 years than non-runners.
As long as there is no injury, repetitive motion may actually protect the joints, perhaps by strengthening nearby muscles.
Similar studies are not available for cycling, though I did find two studies that suggest cycling was associated with a higher risk of knee osteoarthritis.
Considering the health benefits of being active and the fact that obesity is a risk factor for arthritis, most doctors would encourage you to cycle and jog if you enjoy these activities. Unless you have pain or other problems related to cycling or jogging, its unlikely that a doctor would recommend restricting those activities in the hopes of protecting your joints.