So, let's be clear: you have a High Deductible Health Plan (insurance policy) that covers you, your spouse, and your child, while your spouse has an "ordinary" health plan (i.e., not HDHP) that covers ONLY your spouse.
If it helps, there are two things here: your HDHP policy (your health insurance) and your HSA (an account which you can contribute to and pay for medical expenses from). The HSA belongs ONLY to you; it does not "cover" your spouse and child, even though you can pay for their medical expenses from your HSA.
"When I'm asked if I have other health coverage" I am not sure where you are asked this (is it in TurboTax?), but generally it would mean "do YOU have other health coverage?" It is not asking about your spouse's other coverage.
In your situation, you are eligible to contribute to your HSA because you do not have any coverage other than your HDHP, while your spouse - even though covered by an HDHP - is not allowed to contribute to an HSA because she is covered by other health insurance.
Make sense?
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