I don't have any restaurant recommendations, but a couple of cafes worth a mention:
it doesn't have a play area, but in the Niederdorf we like Cafe Henrici - they've always been very tolerant of us when we've been there and have a bookshelf inside with some kids books, crayons and a few toys on. We've also always enjoyed the friendly service (and the Flammkuchen!) there, not a given in Zurich!
There's also " Kafi für Dich " in Kreis 4 next to the Bäckeranlage - though they've been a bit snarly about bringing buggies in to the place when I've been there (don't know what they expect if they encourage families with kids!) and it's hardly been service with a smile, there is a reasonably sized play area at the back with some toys, and a massive chalk board all across one wall. And if the kids get really tetchy there's the big Bäckeranlage park across the road to let off some steam.
We tried Tibits in Basel which was good. Quite a lively atmosphere so I didn't feel like I needed to hush my girls every few minutes. They also have a small kids corner with books, colouring pencils etc..
Also Hiltl and Tibits in Zurich
Zeughauskeller
http://www.zeughauskeller.ch/en/home
Manor and Globus also have good options for kids, if you are tired of the Coop.
Normally if you are eating at lunch, or early dinner, you won't disturb anyone.
http://www.ristoranterosengarten.ch/
Wagamama has a nice corner and table next to it - nice but not great as their carpet and toys have a dirty feel
Migros restaurent in lowenplatz has a small play area inside and bigger one outside. Coop on St.annahof also has a nice one. Haben't been to manor ina while but they used to have a tele in the corner.
Fork and Bottle during the spring and summer is an amazing place to go for brunch with children. They can run around without one being too worried about traffic. Lots of families tend to go there. www.forkandbottle.ch/