sjd, In Dartmouth itself I'd say the most child friendly restaurant is Kendricks in Fairfax Place. They do kids food, are very helpful and I think they have highchairs. Food is a mix: pizzas, burgers etc. but also fish (like seafood kebabs on rosemary sticks with citrus jus, that kind of thing), thai curries etc. and very good chips - don't know what they put on them but they're amazing.
If your children are older, on the pub front The Dartmouth Arms in Bayards Cove is nice. Children aren't allowed inside but there is a small area with benches outside where you can sit and watch the world go by. If your kids are very small though, I wouldn't recommend it as the river is very close with a c30ft drop to it at high tide. Their pizzas are good.
Further out of Dartmouth on the Dartmouth - Totnes road is The Sportmans Arms. It has a childrens room with a pool table (50p) some rubbish toys (so take some of your own), a garden with a swing and a climbing frame and does food although I'm not sure what the quality is like. The menu looked ok when we went though.
At Slapton Sands, The Start Bay Inn at Torcross has a family room and does childrens food. Wouldn't say it's wonderful but it is a kids room!
The nicest pub I've been to with kids is at Frogmore - past Slapton and keep going at the next roundabout. Sorry, but I can't remember what it's called. It has a stream running at the back of it with ducks (safe though, there is a fence IIRC), does great food and has a small childrens room. It's called something like The Bridge? Never can remember but very nice anyway.
The beach cafe at Blackpool Sands does good panini, light meals (burgers etc) and serves wine and beer. Where are you staying? Don't tell me if this is too nosy! Have a great time, HTH.