The good news is that there are several excellent models of trailers out there. The bad news is that well defined guidelines for when a child is ready for a trailer are nonexistent.
So let's deal with the bad news ...
Roughly, you will find a few references that recommend waiting until a child is one year of age. Precisely what that recommendation is based on is vague. From what I can gather, the direct evidence is virtually nil such that it is more of a conservative educated guess. We decided to take Henry for a few short rides at around eight-nine months since he developed good neck control at that age. Moreover, since he was in an aluminum crash cage, we didn't bother with a helmet which would only add more weight to head and make the experience less satisfying. Anecdotally, among cycling enthusiasts, I have read experiences with children from six months to about one year for the starting date. But as I imply above, I would worry little about age instead of the child's physical development and their ability to enjoy the experience. For instance, even at one year, Henry would have a hard time touching the floor of the trailer. He was unable to use his feet and legs to support himself and would slowly slide forward into what appeared to be an uncomfortable slouch. He never complained; but I imagined that it would not be particularly comfortable. I ended up arranging a cardboard box to effectively raise the floor to his feet.
The good news is that there are several excellent models out there. After some investigation, I focused on three brands: Baby Jogger, Chariot, and Burley. There are cheaper models which in truth are fine for a majority of uses. However, we observed lower quality wheels -- the rims would only support low PSI tires for instance --, poor insulation, less comfortable seats, worse storage options, and less features. Nonetheless, I am pretty confident in saying that they are almost certainly just as safe and acceptable for short rides. And assuming that any interested reader will do a little price shopping on his or her own, like bicycle components in general, one will pay a disproportionately large amount of money for an increase in quality/features. Note that I only looked at models with 20" (ERTO 406) wheel as this is the common BMX size. Consequently there are lots of quality tires and tubes available. If you decide to go with a model without suspension, you can find great street tires in this size that will roll well at -- the Schwalbe Big Apple is a famous example -- ridiculously low PSI.
Baby Jogger has an awesome and quick fold. It also converts from one mode to another in a fast fashion. It is pretty expensive and heavy. However, like many products if you are patient, you can find them on sale.
The Chariot has a lot of accessories and it features a suspension. Unlike road/path cyclists where suspension adds little value, in the case of someone sitting down without an ability to use their legs to help absorb impacts, I think that the additional suspension will make a much more comfortable ride for the passenger in the back. Wherever you ride, there are occasional holes, curbs, and so on that will result in big bumps. This helps smooth it all out. Realize that the base carrier does NOT come with the bike hitch accessory which I recall is non trivial. Chariot is carried by REI which has a ridiculously good return policy and offers 20% off coupons that can be used on any single product. With an REI credit card, there is also an additional percentage of the purchase added to your annual rebate.
The Burley D'Lite also has tons of accessories and features a suspension. Unlike the Chariot, it comes with the bike hitch. Consequently, it is less expensive than the Chariot. It used to be carried by REI, but I believe they only carry the Chariot line of trailers now.
If you plan on having a second child as we were, then I suggest going with the double model. There is a weight and width penalty to getting a two child trailer. Although to date, the width of the two-child trailer has not been an issue for me. This is the model -- the double seat Burley D'Lite -- we purchased. I waited until the fall sale, got ~25% off, and still got 5% added to my annual rebate. I know that Burley also offers an infant seat among its accessories which provides more support for babies.
Now I referenced relatively expensive models. From memory, I believe both Chariot and Burley offer less expensive models that have fewer features. For instance, at the time we shopped for trailers, the model one tier down the Burley line lacked a suspension. The good thing about these expensive trailers is that there is a large used market for them. One can often find a used model in Craigslist or recoup a decent percentage of a new purchase down the road. Although before purchasing a used model, do realize that the models have changed over time. For instance, Burley D'Lites from a few years ago do not have a suspension.
Henry and I have done several 30-mile rides together. So far, he seems to really enjoy the rides and is excited when I mention a trailer ride. My rules of thumb are that I try to let him out of the cage every 60-90 minutes where he gets to play, do something interesting, eat with daddy, and so on for 15-20 minutes. That seems to keep the rides lively enough for him to want to do it again. Also ... realize that the damn thing is heavy. The Burley trailer (double) is 33 pounds; plus 28 pounds for Henry; plus whatever supplies I have for the bike, me, and a two-year old; plus the fact that I ride a recumbent which is close to 30 pounds. Picking appropriate gearing and routes makes the experience a lot better. Also, an upgrade that I wholeheartedly recommend if one is sticking with roads and paths is swapping the tires. Not only will rolling resistance decrease but I suspect that the ride becomes more comfortable for the passenger as well. I base this on rides with my recumbent which is designed with a passive suspension. That is, the steel frame is designed to flex. When I hit large bumps I'll hear the trailer hitch and the rear arm make some noise. When I switched to Primo Comets -- virtually zero tread road tires that come in a wide variety of widths -- that sound decreased considerably. If one is patient, Primo Comets can often be found at a discount or used.
Feel free to ask questions and report your own experiences.