Yes you can sue them. Whether or not you will be successful is another question. In a typical negligence suit you must show that the apartment complex had a duty to you, that they breached that duty, that you were harmed in some way either physically or financially and the harm resulted from the breach.
I suggest you contact an attorney if you feel that the harm you suffered justifies the emotional investment you will have to make in pursuing a claim such as this. Good luck!
This response does not create an attorney client relationship nor should the advice be relied upon because it is not specific to your legal situation. Before you depend on legal advice, you should retain competent counsel.
I think this will be a hard case unless there are considerably more facts that support a causal connection between a broken gate and car thefts and other issues.
Let's talk about some legal issues. Negligence is a tort. The advertisement would be more of a contract claim. Those are separate separate legal theories. For now, let's just consider the tort, meaning the negligence claim.
Negligence generally requires a link between the claimed negligent act and the damage. Here, you would have to show that there was a substantial ("but-for") relationship with not having the gate and your car being stolen. That would require more facts about the theft patterns in your area and involve other issues.
As correctly noted by my colleague, you will need to speak with an attorney. Just filing a suit based on the facts you noted may go nowhere at all.
This answer is for informational purposes only and is not legal advice regarding your question and does not establish an attorney-client relationship.