Where should we stay in Dorset?

Swanage fits the bill. There is loads of information on this website and you can read about boat trips from Swanage Pier on this one. It's a lovely small town, the kind of old-fashioned seaside resort that many of us remember from our childhood with plenty of beaches (although discretion is advised at Studland, which includes a naturist area), but it's also in the Purbeck Hills, the Jurassic Coast is just around the corner, there is the railway, Corfe Castle a short drive away (or take the steam train, because there is a station at Corfe). The only problem is fitting everything in to the short time you have available.

For accommodation your best bet is to use Tripadvisor, or that first website I mentioned. I'd guess you will struggle to spend as much as £150 a night, but I could be wrong.

I'd second Kevin's suggestion of Swanage. Other places you could consider are Weymouth and Lyme Regis. Weymouth is an old seaside resort, with a long seafront and unique sand that is so soft they use it for sand sculptures. Lyme Regis is where the first dinosaur fossils were found by Mary - ah, my memory fails me ….anyway, you can still find fossils on the beach sometimes, and your children might enjoy looking.

Thanks Keith, yes Mary Anning.