So far, it might seem like I am leaning towards the latter position, but I do have a ton of sympathy for the former position. However, I will explain why I sympathize with the former position after making a similar example from our textbook and our class discussion. An English museum decided that it was going to "borrow" a piece of the Parthenon for display, but they apparently didn't get permission to take the piece. As a result, the Greeks are still waiting for that piece of the Parthenon to come back into their possession since they have no right to have it. Also (and not from the textbook), many Native American skeletons are put on display in American museums without permission from the specific tribes that they belong to. They also are waiting for the skeletons to come back into possession so as to be able to put them through the proper burial ceremonies. Clearly, the people who are against putting the shipwreck on display would probably be against putting one or both of these items on display because they were taken unlawfully and without proper and standardized wait of obtaining them (just like the shipwreck). I would side with this group on both the Parthenon and Native American skeletons as well because the museums are clearly at fault for taking these items without permission and not giving them back (some might debate on the Native American issue, but they ignore the fact that mourning cannot be fully achieved without proper burial ceremonies...and you need the skeletons for that). However, there is quite a difference between these two situations and the shipwrecked scenario.
The shipwreck near Indonesia was looted by sailors prior to the Smithsonian's being able to do anything about it, and the Indonesian government didn't know how to handle the situation. Therefore, the Smithsonian cannot be blamed in a moral sense for putting them on display, especially if they were not mishandling the items in the first place (...the sailors were...). However, even though I side with putting them on display for historical reasons as well as common sailors being at fault for mishandling the materials, I do think that the Smithsonian is morally obligated to take a step back and try to handle the objects in as much of an official manner as they can. In other words, they need to document what happened to them before they handled them so as to make sure that they don't damage them further, and they need to handle whatever is still in the shipwrecked location with the usual protocol of professionalism (as if they had never been looted in the first place). Putting all the proper steps in place, they would then be allowed to put these items on display. The people have a right to see these items, and it is the Smithsonian's moral obligation to put them on display...but only after they have gone through the proper protocol of doing so (whatever that protocol may be).
The issue here is clearly a global issue, and it gives us a new perspective about our identity as a global and national culture (at least for the people of Indonesia). One event changes everything because - as the Buddha adequately said (along with the scientific world) - everything is interconnected. The whole course of history will change because of this event. Not doing something about it would be a disservice for the world as a whole, and it gives us a new perspective as well.