that's me in the picture above! Hi, all.
I carried both my babies by bike-- they came home in-arms onboard pedicabs, and started in the Yuba rig pictured quite young (baby #1 at 7 weeks, baby #2 at 3.5 weeks).
The American Pediatric Association says babies should not ride on bikes until they are one year old and can support the weight of their head + a helmet while sitting upright in a child seat or trailer. They vaguely cite vibration damage to the spinal cord as reason to refrain from carrying them younger, though if anyone can show me the study that indicates that a baby in a car-seat onboard a bike suffers more jostling than a baby in a stroller or a baby-carrier, I'll give you $50. I've searched, including PubMed and other scholarly databases, and it appears that the APA's finding is based entirely on speculation rather than scientific analysis. Which makes sense, as Dutch infants travel this way all the time without a rash of bike-based brain injuries. I am unconvinced by the APA's recommendation and, for my own children, have chosen to ignore it.
Any form of travel is going to involve some bouncing; once you decide that your baby will benefit more from leaving the house than from being safe inside it you are exposing him to some danger. Auto accidents are the leading cause of death for children in the US, remember. So get over this notion that bike travel is this exceptional form of risk for your baby. We put some foam under the strapped-down car seat to add a bit of extra suspension, and were as careful as possible to choose the smoothest available line while riding, but ultimately, some bouncing is going to happen. And that's probably fine (add caveat here where I am not a doctor nor a lawyer, just a mom with two healthy-and-clever-so-far bike-carried kids, ymmv).
So my feeling is, as long as your kiddo is reclined and secured, even the littlest babies are safe on bikes, as long as the bikes have competent riders and safe places to ride.
Nota bene: a baby in a car seat should not wear a helmet: the extra material under the back of their head would put unsafe strain on their neck and defeat the purpose of the reclined & secured position. This means that, in states with a child (or universal) helmet law, carrying a baby on a bike is de facto illegal. I feel that strapping a baby to a big seat-shaped piece of styrofoam rather than a little head-shaped piece of styrofoam provides similar protection, however, the letter of the law in my state (and maybe yours) disagrees. I'm grateful to have never had to take the issue to court, and haven't heard of anyone else doing so either. But, something to keep in mind.
For another perspective on this issue, see: http://totcycle.com/blog/baby-on-a-bicycle.html
Also, the SF Bike Coalition's Family Biking Guide has a great section on biking with infants: http://www.sfbike.org/?family_guide