Can we claim Gift Aid if one half of a couple pays the membership fee for both people?

If you offer a reduced membership fee for couples, then this is essentially like a family membership - and so you can claim Gift Aid as long as the normal conditions for claiming Gift Aid on membership fees are met. The person who makes the payment must be the person who has given a Gift Aid declaration to the charity. 

If they are paying 2 lots of full price fees – but just one person is making the payment, then the fee that relates to the person making the payment can have gift claimed on it (assuming the normal conditions are met). But the fee relating to the other person could not be claimed  

If the individuals made two sperate payments, you could claim on both (assuming normal conditions are met and they are both UK tax payers)