If you run with a Chariot or other brand of bike trailer turned jogger, protection is built-in, but if you don't, I highly recommend investing in a raincover ($50) to protect your child from the elements and rocks that fly up. If you're concerned about UV rays, a sun shield ($69) offers ultimate protection, or be sure to put sunscreen on both you and your child before heading out.
Another way to keep a baby safe while running is to use a tether — a strap that attaches from your wrist to the stroller, just in case you accidentally let go. Bring your phone too, so you'll be able to call for help if need be. I also use my phone to track my workout and to play music on speakerphone. That way it entertains us both and I can still hear my daughter if she needs me.
If you're also running with your dog, do not tie the leash to the stroller. Hold the leash in your hand so when Fido sees a squirrel and starts to sprint, he won't take the stroller with him.