It will be a problem.
If you signed, you committed and unless the LL fails to deliver possession on time, you are stuck with your decision.
You have to determine whether it was an isolated incident or whether the apartments are in a crime ridden area, whether you are willing to face a suit by the LL (LL may or may not sue you, depending on their circumstances, policies, etc) for breaching the contract.
You may consider speaking with the LL and explaining your reluctance, your safety concerns, the lack of complex access locks, broken doors, poor lighting (if any of these matters apply). The LL MAY let you out of your lease based on your concerns if LL believes he/she can easily find another tenant, or he/she believes you are going to be a thorn in the LL's side because of the expenses associated with the safety matters you are raising--but LL is NOT obligated to let you out of your lease--they'd be doing you a favor--keep the conversation polite.
Also, you may consider hiring a lawyer to negotiate on your behalf. Might be worth a few dollars to see if that lawyer can find any grounds in your lease agreement, or with the complex facilities that may open the door for you.
Finally, murders can and do happen anywhere--it may have been an isolated incident--talk to other current residents to get a sense for the environment and whether you could expect to comfortably and safely stay there.
READ THIS BEFORE CALLING OR EMAILING ME: I am licensed to practice before the state and federal courts in Virginia. Addressing your issue does not create an attorney-client relationship and I AM providing you educational information NOT legal advice. You should speak with an attorney to whom you have provided all the facts, before you take steps that may impact your legal rights. I am not obligated to answer subsequent emails or phone calls unless you have hired me. I wish you the best of luck with your situation.