Damage and Liability
Some mobile office companies offer insurance coverage directly for their rental trailers that can protect you from third-party claims for property damage and bodily injury. Other companies offer you a damage waiver, which will cover damages to the rental unit or its attachments beyond a certain minimum amount. Please note that every supplier is different when it comes to insurance coverage and you will want to verify what is offered with each company before making a final commitment.
You can also take out your own insurance policy by contacting your insurance company to see what your options are.
Theft
Theft can always be a problem anytime there is a building that is unoccupied both during the day or night. The best way to ensure your stuff is safe is to invest in a high quality lock and don't give out the code or key to anyone you don't trust.  A lot of companies will offer you a high-end locking mechanisms at an additional cost.
If your mobile office trailer has windows, you could also consider adding metal bars to deter theft. A lot of companies use their mobile trailers to store their valuable equipment, machines, and supplies on their job site. This can be a huge financial loss for the business if these items are stolen both to replace the equipment and to cover the time lost on the job and having to deal with the hassle.
Be sure to check and see what insurance options are available that cover theft if you are in an area where you feel this could be a problem that you will have to deal with.