Am I liable if I made a rental payment to a drop box but the box was broken into and the payment was stolen?

Legally, IF you placed the payment in the drop box, at that point, you discharged your obligation and you are not responsible for what happens later, including if it is stolen out of the drop box, any more than you'd be responsible if the landlord's bookkeeper embezzled the money. Once a tenant pays, he or she has paid, and it does not matter what then happens to the money.
Practially, how will you prove you made the payment, if the landlord claims you did not? If the landlord wants to you to re-pay and you do not, and tries to evict you for nonpayment, it will essentially come down to who the judge believes--you or the landlord. If there was a police report or other evidence that the box was broken into, that bolsters or supports your story but if you have a history of having paid late or failed to pay on time, that will hurt you in court--it will make it look more reasonable that you are just taking advantage of the box being broken into to claim your rent was stolen when you never paid it. 
If you paid by check or money order, put a stop to it. If you can stop it before it is cashed, then simply re-issue a new one to the landlord--that's the best way to deal with this situation.