We lived in Streatham for nearly 10 very happy years. We've since moved to be nearer elderly family etc.
In terms of family life, I'd recommend anywhere around Streatham Common - large green space, fantastic playground, the wonderful Rookery, lovely cafe at the top of the hill, some good places to eat etc. Near the big supermarkets too. Streatham Hill is a bit smarter, but you don't have the open spaces there.
On the plus side, you get a hell of a lot more for your money than you do in most other parts of south-west London, and there's an extraordinary community spirit: welcoming of everyone and keen to include them in whatever's going on. Residents are rightly v proud of Streatham and it being such a mixed community in every way. I've never lived anywhere where my neighbours were so friendly.
On the down side, and one of the reasons we moved, is that because it's still a bit rough around the edges, I didn't always feel hugely safe wandering around in the evenings, and as my three girls are teenagers/nearly teenagers, I was apprehensive about them going out and about too. I may well get shouted down for this, and I realise that there are no really safe areas of London, and we certainly lived there for a decade without anything happening to us personally, but we certainly knew of friends / friends' children having less than pleasant experiences.