The Handlebar and Brakes
The first thing to say is, make sure your handlebar is a comfortable height! Have you ever gone for a run holding your arms above your head? No? Good, because that would be crazy. But you can imagine how uncomfortable it would be, right?
Most good strollers have adjustable handlebars that can be set to multiple different heights, for even the tallest and shortest amongst us.
You're also going to want a handlebar with a decent amount of padding. This doesn't matter so much when you're just walking around town, but when you're running you'll come to appreciate that extra layer of protection on your hands. Trust me, after a long run pushing two babies in a 30lb stroller, your hands will be feeling it enough!
Whilst we're talking about handlebars, it's also worth discussing brakes. Seems like a weird segue, but a lot of jogging strollers have hand brakes that are mounted on or near the handlebar, as well as a footbrake.
If you can, it's worth buying a stroller with an easily accessible hand brake, because they are so much easier to use when you're moving than foot brakes, and you only need one loose rock or slippery big of ground for accidents to happen. If it ever does, you'll be happy you had your brakes in easy reach.
Like on cars, the best types of brakes are drum brakes. What this means is the brake clamps down all the way around the axle, putting force on every part and stopping you faster. Again, these are worth looking out for, especially if you regularly run on rough or loose ground.