Sure you can sue - anyone can, for anything. Whether you have a valid claim worth pursuing is a different issue. The first question is what your written agreement provides as far as security, or whether it provides there is no security and you waive liability. As far as the signs, the signs themselves have become an accepted practice of deterrence. Obviously, they don't prevent all crime but they may prevent some. That is not "advertising." Did you ask about security before renting? If you claim it was an important factor, be ready to answer that question. Beyond the legal issues, what did you lose and is it worth thousands in legal fees (or a hefty contingency fee) and your time to pursue? You MIGHT have some chance at a claim depending on your contract and other details.
I agree with Mr. Riddle that you need to read your documents. Often, these places will limit their legal liability or liability for damages to the amount of rent you paid for example. So the first think is look at your documents.
I do think that you may have a claim for false advertising and possibly for negligence in leaving he gate open, but will the damages you can recover make the case worthwhile?