To open or contribute to an HSA you must meet certain eligibility requirements. Specifically:

You must be covered by a qualified high deductible health plan on the first day of the month
You cannot be covered by any other health plan that is not a qualified high deductible health plan, including spouse’s health insurance
You cannot be covered [...]

In Administrative, Procedural, and Miscellaneous Notice 2004-50, the IRS addresses this question. See Questions 2 and 4, excerpted below:
 Q-2. May an otherwise eligible individual who is eligible for Medicare, but not enrolled in Medicare Part A or Part B, contribute to an HSA?
 A-2. Yes. Section 223(b)(7) states that an individual ceases to be an eligible individual [...]

The account holder in this scenario has family coverage and is therefore eligible to contribute the family limit . They may also contribute the catch-up amount, if he or she is age 55.
The spouse is an “ineligible individual” in the eyes of the IRS. The IRS allows married couples to assign the tax deduction to each spouse [...]