I have recently purchased my first pop up, a 2009 Jayco Jay Series 1206 and was wondering what the group thought about adding a stabilizer and if so, which one do you recommend?
Are you sure you don't have stabilizers already? I was under the impression that they came with them preinstalled from the factory although yours may have been removed over the years. In any case, I would go back with the same ones again although it doesn't have to be the same exact model and manufacture, just stay with this same design though as shown in the image below. ~CA
A thought occurred to me that perhaps you are asking about something other than the 4 "stabilizers" shown in each corner, if so feel free to expand on your question.
As Craig said above, most PUs already have stabilizers. I've owned three in the past and all of them had 'em. Now if you're referring to a sway control, then it depends on the model. In my experience, any of them with an extension (pop out) on one side tend to be a bit squirmy while towing (especially in the rain). It's easily solved by adding a friction based sway bar (e.g., https://www.etrailer.com/Accessories-and-Parts/Reese/83660.html?feed=npn&gad_campaignid=1672411894). If you can identify exactly what you're referring to, we can likely offer a more exact answer.
In 2001 we bought a new Coleman Mesa popup, and towed it with a 1/2 ton Silverado 4x4. It was one of the heavier pup they made, and by the time swmbo got done filling it up with with all her "needed" stuff, I was left with sway issues. I promptly added a friction sway control, and problem solved! You could try moving weight around, but the sway control can't be beat!! Good luck!