I’ve been using one of these for our little ‘un since he was about 9 months. Has been fine both on and off road.
http://www.wiggle.co.uk/p/cycle/7/OK_Baby_Orion_Child_Seat/5360032134/
He loves it – I’ve put a little bell on the handle so that he can ring it himself,
We ride local bridleways regularly but also done things like the “Afon” family trail at Coed Y Brenin without any problems.
It feels really safe, as he is protected between your arms. Contrary to appearances (and what some folk on here will tell you), the weight is also in the middle of the bike, so you’d have to do something really daft to endo.
If you’re riding off road a lot I would recommend using some thread lock on the clamps though just to be on the safe side.