How old are the kids? Brixton Village isn't great with babies (as much discussed on another thread) due to lack of toilet facilities. There is a toilet but it's outside the market, so you will need to leave the market and walk to it to use it - there is supposed to be baby changing there but I've never seen it open. The nearest option would be The Ritzy cinema, Morley's Dept store or Brixton Rec for actual baby changing. Tons of choice for eating though.
Sitifis is good but v. small. Good for babies - no baby changing per se but you can just about manage in the toilet (just). Lovely lovely mezzes for not much money. Really nice owner. Small box of toys for babies. My daughter loves it there.
If they're a bit older The Duke of Edinburgh has a decent sized garden as does The Grand Union on Acre Lane. I've not been to the former for AGES but I hear it's gone quite gastro pubby compared to previous incarnations. The latter is a chain burger place - only been there the once myself since it became the burger place, again AGES ago - it's not my cup of tea. Tbh.. last time I went to either was pre-baby but I mention them because of the large gardens.
The Ritzy would be ok too depending on the time of day. Fairly varied menu, service can be a bit erratic though.
The Lounge is popular also - has baby changing. Not sure if they have high chairs though - last time I went ours was too young for a high chair. Lots of sharing plates/burgers etc. Decent size for a group.
Brazas, near the Hootenanny is very kid friendly - again, no baby changing but the toilet copes (again just). They always bring out little presents for the babies - strawberries, babyccinos etc. Nice varied menus, everything from omelettes to burgers to fishcake type stuff.
Mango Landin' is good for kids but only really open during the daytime at weekends. Nice varied menu.
Herne Hill is really the place to go with kids - Cafe Provencal has a whole section dedicated to parents with kids - huge boxes of toys, books, children's drawings and baby changing - sofas, tables etc. Food varies from breakfasts to proper meals and burgers etc
The Florence has a separate playroom out the back of the pub adjacent to the (small) pub garden with tons of toys and TVs, baby changing etc... gast.ro pub
Pullens welcomes children (gastro cafe) as does The Commercial Pub (small garden) and there's also a Pizza Express.
The Lido cafe is very popular with mums and kids, but it can get very crowded. Gastro style... also obviously by the pool if there are older kids who like to swim.
I don't like all of these places personally but they are the ones that immediately spring to mind as being popular - all for different reasons - it depends really on how old the kids are I think.
There's probably a few I've not thought of...