Hi Carol

We spent 2 weeks near Cavtat in June 2006. We stayed at the Albatross Hotel about 15 mins walk from the village. I do not remember there being any public swimming pools. The larger hotels have pools for residents. In the village itself there is not a beach as such, just some rocky areas you can swim from. There is a beach in front of the Albatross and its sister hotel. It is a mixture of pebbles and sand but we were able to swim from it. We really enjoyed the area with easy access to Dubrovnik etc but we are a 50 year old couple. It may not be the best area to go to if you have young children. I think you may need to be further north maybe like Makarska for a beach. Perhaps you could fly to Split instead.
Having said that we saved a lot of money by booking the Albatross direct on the Iberostar Hotels website - the downside being you have to pay when you book. There was stuff going on for children though in multiple languages. I don't know if your budget would stretch to this. Try looking on Google Earth and you will see the two big hotels.
The good thing about Cavtat is that you are in your accommodation about 30 mins after the plane has landed. Just look on the Dubrovnik airport website for a link to pre-book your taxi. From what I remember it was about £8 each way.

If I can be of any further help please PM me or ask away on here.

Cheers, Jean