I take the opposite view.
I think it's more that when wealthy people use/abuse drugs, they have a way bigger financial, social, and medical safety net, so that their drug use ends up having a far smaller (and less publicly noticeable) impact on their lives and future opportunities.
Yes, people with less access to financial security have more statistical drug use, but the statistics might also be skewed by
1. the fact that non-wealthy people have more interaction with State-funded services (police, rehab, therapy, prisons), which track and report drug use more systematically & robustly than the obscenely-expensive, private, and very discreet rehab facilities the wealthy have access to
2. because of the aforementioned safety net, there's simply less likelihood of a wealthy person being 'caught' using drugs by an entity that would report those statistics (police, schools, etc).
The higher one's poverty rate, the more vulnerable one is to having their actions scrutinized, catalogued, and punished by the State.