I think they've replaced all the coast path that fell into the sea round Lulworth/Durdle, but might be worth checking.
Lots of walking - Golden Cap and so on. Please pay attention to any warnings about diverted paths and so on - bits of cliff have fallen off quite badly (and sadly fatally in one case) in the last couple of years.
Most of the NT stuff down west Dorset is coastal/landscape stuff like Pilsdon Pen (great views if it's clear, but steep), where you don't have to pay - Corfe and Kingston Lacey are the main payable ones, but they're east Dorset.
If the weather's good, then you've a pick of beaches, though not all the ones along Chesil Beach are so good for just sitting (bit pebbly - you can tell whereabouts along the beach you are by the size of the shingle, they say. Can't say I can...) But there are some nice places like Eype. And you have to do some fossil hunting round that way. Well, actually, there are lots of good fossil areas - it's not the Jurassic Coast world heritage site for nothing.
Abbotsbury has the children's farm as well as the swannery. There's loads to do in Weymouth for children.
If you go west from Lyme and cross the border into Devon, there's the tramway at Seaton.
You can always go and run round Maiden Castle at Dorchester, and if the children need to burn off energy, make them do it the hard way, by running up and down the ramparts like invading Roman legionaries.
(Feeling a bit homesick now.)