Elon Musk and his supporters offer several explanations for his contrarian views, unfiltered speech and provocative antics. They’re an expression of his creativity. Or the result of his mental-health challenges. Or fallout from his stress, or sleep deprivation. In recent years, some executives and board members at his companies and others close to the billionaire have developed a persistent concern that there is another component driving his behavior: his use of drugs. And they fear the Tesla TSLA -0.18%decrease; red down pointing triangle and SpaceX chief executive’s drug use could have major consequences not just for his health, but also...