The American legal system allows you to sue just about anybody you want. What you are asking is whether your suit would have merit, and not be dismissed on pretrial motion.
There are a couple theories at play. The place to start is with your leasing contract. Is there language in the contract that talks about secure parking or gives some indication of the security of the parking garage? Most likely there isn't. But bringing a suit on contract is always most often where you start, because contract is in writing and provides certainty. Judges and juries want certainty. If the contract says we guarantee secure parking and your car was stolen you have contractual language to stand on.
Failing that, you fall to the more encompassing legal theory of tort. Tort law most often means negligence; that someone should have provided a duty of care but failed to do so. In your case, basic measures that should have been taken to secure your parking garage were not taken. For example, doors were not left locked, or the the garage doors did not function properly and were left open for long periods of time.
But having a good legal theory and actually taking steps to file a suit are totally different. Taking a legal action is expensive and time consuming. This is where insurance comes into play. Did you have insurance on your car? Does it cover theft? These are important because the costs you would incur to take legal action against the apartment complex could easily exceed the cost of buying a new car.