This question gets asked a lot. And the long and short answer for this one is NO! Absolutely not! And this isn’t my opinion. There have been many research studies done to prove this. The results from the legendary Framingham study- an intensive medical research project that followed its subjects for over 40 years tells us so. It studied the same people (584 men and 820 women) for a long time and concluded that no association was shown between the level of physical activity during middle age and the occurrence of knee osteoarthritis in elderly persons. 

In layman’s terms: Today’s exercise will not cause tomorrow’s pain in the form of arthritis. So provided you are training correctly and have proper running shoes, you have nothing to worry about.