Eames molded plastic rocker: Stable?  

I
don't have kids, but would imagine the potential for injury exists with any chair. .but a rocking chair might just be a more attractive playground ?..

Not a good option
Fine for a living room or bedroom maybe. But I'd never put one in a kid's room. Mine seem to find them more like toys than furniture. Want to stand up backwards whilst rocking, etc...

Hmmm
I wouldn't think that it would be any less stable or more prone to tipping over than any other rocker, unless it was one that was very heavy duty.

Not necessarily unstable, woody...
That wasn't my point. Point being I'd never leave my kids alone with a rocking chair unless it were a heavy duty chair or glider as you say. Even the nice Danish high back rockers aren't safe, for the kid or the chair. And the playful lure of an Eames RAR only invites more playful, aka destructive behavior towards the chair and possibly destructive results for the little weasels themselves.

As a kid
I used to pretend my parents' rocking chair was a surfboard... Yeah, they were none too pleased about that.

The stability's not so bad
I'd be more worried about the wire base. All those openings are just begging to have a child's arms / legs / head ensnared in them. Ironic for something HM presented as a "baby rocker."
Also, be aware that the repro bases often have child-unfriendly sharp wire ends and welds.

?
Assorted scars and/or missing teeth add charm to a child...as long as such a wound happened in an acceptable manner. Loosing an eye whilst over-rocking an Eames chair would certainly add character to the little weasels plump mug. For my own home, I keep a steady supply of duct tape on hand for when the little one's come a visitin'. Just throw a helmut on the little squirt, and let him rock till the chair melts..for gods sake!
http://no kids no link no rocking chair no worries no kidding

Well...
our little one has not yet had any close calls with ours and has never attempted to stick her head through the truss-work. Oddly enough, when she "over-rocks" it just causes the chair to scoot backwards as opposed to toppling over.
But, as was said earlier, it could prove a little more precarious if she decided to STAND in it...but that could be said for MOST chairs.