I have a trailer with suspension, but I don't believe it has provided much benefit while cycling. Mine is multi-use (jogger, stroller, ski) and I have found it has provided some benefit while hiking.
On the cycling side of things, most of the time I am pulling the trailer with a rigid bike (no suspension). I am riding with 5 weight bearing contact points while my kiddos are riding with a full body contact area and no need to maintain control (which is what suspension is designed for). When they were younger, my trailer had an infant sling that kept them fully suspended (similar to a hammock). Now that they are old enough and large enough, they sit in the trailer proper and receive what I perceive to be very little benefit from the suspension while I receive the extra benefit of pulling it's weight.
I have been very grateful for the suspension on a few hiking trips when taking them down a rock garden. However, the vast majority of the time, cycling, I'd prefer to not be towing the extra weight.