The Financial Times asks the right question, and offers several useful answers.
A couple of caveats:
High-resolution recordings of soon-to-be-pulverized monuments may well be the best one can do in the absence of military action to secure sites or to deter movement of looters and iconoclasts onto sites. It should be clear, however, that what we lose when we are left with mere records rather than the things themselves is not just the materiality of the things but the knowledge that materiality may hold. A hi-res photograph captures only the visible and only certain aspects of the visible. It is much better than nothing but not a perfect substitute, regardless of whether one values authenticity. Craftspeople already know well how to make fakes that can fool even some experts -- one archaeologist has suggested that something like a quarter of the Meso-American artifacts in museums are probably fakes. But that's not a cheering thought.
Preservation, new or otherwise, of already-excavated monuments and artifacts from iconoclasm or other war-related destruction is a different problem from that of preserving (the context of) not-yet-excavated artifacts from looters. The article notes that what we need to do is to "stall the economic benefits of looting", and suggests that looting can be reduced by better policing of the international antiquities market. That is absolutely correct. The question, however, is how policing can be improved. ICOM's online guide to the types of artifacts that emerge points to one answer: more information. Police need more information, and not just to be able to seize or refuse import of banned items (which ICOM's guide can in some cases assist in), but to go after the smuggling networks by tracking the chain of ownership of seized antiquities. As things stand, dealers and auction houses buy and sell antiquities without disclosure requirements. That has to change. There should be transparency regulations requiring any sale of antiquities over a certain threshold price to be reported with the name and address of the buyer and seller along with photos and other identifying information. (The newly released app from ARCH shows how such reporting might be done in a very streamlined way.)
But even with such information, policing is going to be ineffective absent the funding resources needed to pay for site guards, customs officials, antiquities policing units, and local investigations to do the job of monitoring and going after the criminals. And despite professed new interest in cracking down noted in the article, it is very unlikely that governments in general are going to devote scarce resources to this problem. As the article also notes, the opposite is the case: heritage protection is being outsourced. And foundations and NGOs are unlikely to pick up the slack, especially given that policing, unlike restoration, is a permanent task. Some suggest that looters could be deterred by moral suasion from local religious or traditional authorities, and that undoubtedly must play a role; we know for instance that Muqtada al Sadr was able to turn looting on and off by issuing fatwas. But there are too many sites and not enough moral suasion for this to be the only solution. Another suggestion, being pursued with some success by the Sustainable Preservation Initiative, is that locals will guard their own stuff if they have an economic incentive for protecting it as a revenue-generating tourist attraction -- but while that may protect already-excavated individual sites it won't preclude those locals from going off to dig the unexcavated ones. What's needed is a sustainable revenue stream to pay for more and better policing of unexcavated sites.
The funding solution, as I have suggested in the past, is to tax the higher-end licit antiquities market, with proceeds going into a fund for international heritage protection.