Must agree with other respondees that Brockwell Park is the place to go and the new playground facilities make Clapham Common's play areas look distinctly second rate.
Our two year old does enjoy the paddling pool on Clapham Common however (opposite Holy Trinity Church?) although it gets a bit boisterous at times and there are usually some "interesting" things in the water and be warned, it is extremely slippy as I have found to my cost.
Yep get to Brockwell Park...the best park in south London and so much more interesting to explore than Clapham/Wandsworth Commons.
Happy splashing