I have a dependent adult child living with me with no income and no health insurance. Am I responsible for his health insurance or is there a waiver somehow?

[TaxGuyBill]

If you claim him as a dependent, you are responsible for his health insurance, including the penalty for not having insurance.

Although nothing in your post indicates you qualify for a waiver for the penalty, you can check here if something seems to apply to you:

https://www.healthcare.gov/health-coverage-exemptions/forms-how-to-apply/