Are both of you enrolled in Medicare? Beginning the first month you enroll in Medicare, you cannot contribute to an HSA.
If you both would otherwise qualify to contribute to an HSA (covered under a high deductible health plan, no other health care coverage, not a dependent on someone else's return), and just turned 65 this year, you can contribute to an HSA but your contribution is limited.
Example:
You turned age 65 in July 2016 and enrolled in Medicare. You had an HDHP with self-only coverage and are eligible for an additional contribution of $1,000. Your contribution limit is $2,175 ($4,350 × 6 ÷ 12).
https://www.irs.gov/publications/p969/ar02.html
Be aware, there may be late enrollment penalties for not enrolling in Medicare when you are first eligible.
