Definitely seek out a qualified piercer (not one of those bodgy franchise places like Cosmetics Plus, either - their piercers are given basic sterilisation training and not a whole lot else, not to mention the quality of their jewellery is terrible - go for a tattooist/stand-alone piercing studio with a piercer who knows their stuff), and one that uses needles over guns.
The reason guns are not ideal, is because the back of the earring (which does the actual piercing) is blunt, causing trauma to the tissue, and the spring-loaded gun cannot be properly controlled by the person handling it, so there is the risk of it catching, not firing properly (leading to a half-pierced earlobe), and the guns cannot be properly sterilised so there IS a slight risk of infection. A piercing done with a needle is sharp (meaning a 'clean' piercing with less risk of infection or cysts caused by tissue trauma), able to be controlled and maneuvered properly by the piercer, and while it takes a little longer, it is definitely safer, less painful and there is less risk of infection or complication.

That being said, I had my DD1's ears pierced with guns in a hairdressing salon (she was 15 months old), and she did not have complications - still, on a baby so young, I would definitely be trying to eliminate as many risks as possible. You may be hard-pressed to find a piercer who is willing to work with a child so young (they may wriggle around and cry etc), so you might have to just wait until she's older