How can I control who can add money to the gift collection?

It's very simple. From your Thankbox management page just click the Manage gift collection button from the side menu.

On the next screen you'll see the gift collection controls with the total amount collected. There you can change who is allowed to donate money towards the collection and see it - either just you, as the organiser, or everyone who can leave a message. You also have the option to remove the collection if you no longer want to collect for a gift.