What do you suggest for a drip tray for an indoor display?

To make a drip tray you need a larger bowl to catch the melting water and a way to elevate the globe so it does not sit in the melting water (see detailed drawing).

When making centerpieces it is best to keep all the action as low as possible so people can see over the arrangement. That said, I would use a larger low-laying dish with edges that lip up a few inches. Those kind of dishes can hold a surprising amount of water. Then I would find a short tiered cake plate to put in the middle of the low-laying dish. Then put flowers or other decorations in the rest of the low-laying bowl to create an arrangement.

Keep in mind that everything you put into the bowl decreases the amount of water it can hold. I usually fill a Wintercraft balloon to the size that I want the globe to be and then empty the water into the bowl I want to use. If it does not overflow, its good to go.