Regardless of where you're considering buying a home - whether it's downtown, southside, West Chatham, etc. - I strongly encourage you to investigate the neighborhood of interest. The best way to do this, as others have stated, is by checking with the local police department here in Savannah. You can look online as well. Many popular sites such as Yahoo! have information that you might find helpful. Crime can happen anywhere, and using good judgment and listening to your gut instinct is always smart.
The police department can provide actual crime statistics for different blocks, and these are published in the Savannah Morning News each Monday.
I have lived Downtown for 20 years and have raised a family here, and all my real estate transactions are in the Historic Districts, so I feel confident in giving you this information. Is there petty crime that needs more attention - yes! Would I live anywhere else in this area - absolutely not, it's fantastic living downtwon. Hope this helps.
You aren't the first person I've encountered that asks this question. I always tell people 2 things...
1.) You can go to the Police department in the area and speak with them about crime statistics and reporting, I since I am not in a position to accurately report on or quantify that question.
2.) I have lived in Savannah my whole life, in various areas of the city including the historic district, and never had a single problem. I never put myself in a position where I would be unsafe, no matter what neighborhood, or city I am in.
Here is a website that I like to look at now and then... but if this is something that truly concerns you, take a trip to the police station.