My ex did a lot of research on this as back then, we were together and we wanted to get active while having our baby along, who was quite young, like yours. He said the trailers are safer than bike seats because if your bike hits a rock or other obstacle, it can cause some real whiplash (if the baby is lucky enough to have not flown off the seat by then). He researched a lot and found that a trailer is the best option, especially for an infant, and he found a trailer that had very good reviews and could also double as a stroller. It was completely covered from the sun, rain, and the plastic windows could be rolled up in case it wasn't sunny or rainy and it'd have mosquito netting so she wouldn't get bitten. It had a sling seat that could fit her and a friend or a doll and had a 5-point harness for extra security, like the infant car seats do. It folded completely and had a non-tipping mechanism which made it safe as well as convenient. It is made by Schwinn. Unfortunately, once we separated, he kept it, so I can no longer take her for rides and he stopped exercising, so he left it in the yard of his house, unattended, for over 2 years. Last time I saw it, I almost cried as it was full of mold and nothing like what it used to be and had he not been so greedy with it, it would have been used and kept in good care by me, but anyway... It was pricey; we paid over $200 but I found it cheaper in case you're interested: http://www.babyage.com/products/13-sc760_schwinn_schwinn_...
I think a seat is more of an option for a child who is able to hold its neck upright without assistance and is a bit stronger, in case of an accident. The man at the bike shop recommended the bike seats after a child is about 2, or between 2-3. Before then, he said he recommended the trailer, just like my ex had found out through his research. Something to be aware of though, these trailers, although lightweight, can be a bit heavy if you're out of shape (like I was) and you have to add the weight of the child to the weight of the trailer when taking the total weight you're pulling into account. If you're not used to hauling weight, you may have a harder time pedaling and you may want to build up on your strength before pedaling with the trailer or you will find yourself tired in no time, unless you enjoy the extra workout because you're already in shape. Good luck and hope I was of some help!