@Rebecca1219 Unfortunately theft is an unavoidable risk that comes with giving unattended strangers access to your home. Keeping a locked cupboard is one way that hosts attempt to mitigate this, but as you've experienced, a lock is no deterrent to guests with criminal intentions.
Most petty thefts go unreported or unsolved, and the perpetrators tend to get away with it. But you're not entirely without recourse here, minimal though it is. You can file a police report; it's unlikely to lead to the recovery of the sculpture but it gives you documentation. You may try sending this along with pictures of the broken lock to Airbnb and ask that they investigate this guest (who may have a pattern of similar incidents). You can attempt a claim for the damage to the cupboard, though the odds of success are not great. And of course your review of the guest should send a very strong signal that the guests are untrustworthy and you do not recommend them.
On the tougher question of how to reduce the odds of getting guests with criminal intent, some steps you may already be taking. Although Airbnb likes to scare hosts out of declining requests, you can screen guests by not using Instant Book, check that profiles have verified ID before accepting, include among your rules that guests must present ID upon arrival, and set both a minimum stay of more than 1 night and a maximum stay low enough to avoid your penthouse being treated as a residence.
Generally, though, your kind of listing is not one that I recommend having on Airbnb. With a home furnished with valuable antiques, you need to be able to hold and control a real deposit - which is not possible on this platform. Also, your texts place heavy emphasis on opulence and glamour, which is rather embarrassing to people with real money but very attractive to pretentious thieves and con artists. Airbnb is a high-risk operation with greater rewards for those who furnish with Ikea and Craigslist..
