Lots of little enclaves in Streatham have a good community feel - especially if close to a school with a tight catchment - look at Hitherfield, St Julian's (Streatham / W Norwood borders) or Streatham Wells. (I know less about the ones further South. Is Granton in Streatham?) But check - proximity can sometimes need to be v close.
Streatham is a huge area - transport is v good - if you are in the streatham Hill area you have a choice of Streatham Hill overground or down to Tulse Hill for the Thameslink, with a v quick connection to the excellent polka theatre, for instance. Short distance from some v good big parks - as mentioned by Serenity - I like brockwell, too, and there are v nice small neighbourhood parks - e.g Hillside gardens, which has a one o'clock club, also Unigate Woods. There is a playgroup and active toddlers club at St Margarets Chruch in Streatham Hill, and also I believe at the big Church near the Sainsbury's up the other end of Stretham. Depending on where you live, it can be within close reach (even walking) to Tooting Lido or Brockwell lido.
Cafes - well, the High Rd is horrible (has a good selection of shops but is 6 lanes of A23), but Pizza express does well as child-friendly, as does Perfect Blend in the day - I have seen lots of Mums and babes / toddlers in there.
I think it depends a lot on the group oif streets you are in and immediate locality - in our cololection of roads everyoine says hello and all the parents know each other. Look at schools ofsteds, etc. But, as a recent convert to Streatham (from Brixton) I would say that it still represents good value compared to similiar areas, and yes, is a good place to bring up kids. I live at the Streatham Hill end.
Not sure where the Stretham picture house is....but there is a big Odeon in the High Rd.