Last time I was there(2009), they had security screening similar to a commercial airport. As mentioned, it is private property, so obviously they make their own rules... and according to GA code, it is not illegal per se (only a misdemeanor if you refuse to leave once asked), but they will not allow you in once they find out you are carrying through the metal detectors.
I used to live less than three blocks from there. Last time I went, I literally walked from my apartment in less than ten minutes, so I am very familiar with the area... and it is a bad area. There are several homeless shelters within a few blocks and section 8 housing is mixed in with the Centennial Olympic Apartments directly adjacent to it. It is downtown, it has "low-income" housing nearby, homeless shelters nearby... so, unfortunately muggings are relatively commonplace. There have been several carjackings, motor vehicle thefts, and murders as well since I've lived around here the last 5 years. They view people from out of town as suckers who can be hassled into giving them money, robbed or worse.
Personally, I do not go anywhere around here without my pistol. But at the same time, leaving it in the car in that area would be a HUGE mistake if you use public parking. My car has been broken into twice in the last five years here.
My suggestion would be this: Take your pistol, PAY FOR AQUARIUM PARKING, and leave it in the car... It will be expensive. Maybe even $20 or more to park, but if you are going with your family and need your pistol, then it will be worth it to have secured parking if your pistol stays in the car. Carry while in route, and put it back on as soon as you get back to the car. It's unlikely that you'll find trouble, but still much more likely than any other place in the Southeast so act accordingly. Don't be a sucker to the street people and con artists either.
Oh, and if you do take my advice and pay for aquarium parking, it is attached directly to the aquarium itself. So there will be no walking for blocks downtown while disarmed. And yes, you will see future America's Most Wanted prospects while inside and they may crowd you and be completely rude, but you should be alright.
P.S. Keep your wallet in your front pocket.