You might prefer Port Douglas to Cairns. In PD you'll be closer to the reef for snorkel tours and closer to rainforest sites such as Mossman Gorge and Daintree National Park, both of which are well worth a visit. But the best thing about PD is Four Mile Beach which is right in town. (Since it will be stinger season, you'll have to restrict your swimming to the netted area, but you can walk all up and down the beach which is great.) The last time were were in PD for 6 days, we did 2 snorkel trips (one of which was with Wavelength, our favorite), a day at Daintree NP, a morning at the Wildlife Habitat for "Breakfast with the Birds" and an afternoon at Mossman Gorge, a day to Kuranda (train/Skyrail combo or Skyrail both ways if you prefer) and a day of relaxation on the beach.
If you like to sail, plan to be at the Port Douglas Yacht Club on Wednesday afternoon where visitors are invited to sail for free. http://www.portdouglasyachtclub.com.au/

I'm always mystified when its claimed that Port Douglas is closer to reef than Cairns, having taken our own boat out from both ports have never found that to be the case - much of a muchness depending on what part of the reef visited.

For a snorkelling sailing family, how about a liveaboard, such as Vagabond - a 65' cruiser yacht which only takes 11 passengers. www.vagabond-dive.com Vagabond is out of Cairns, skipper has roaming reef permit and picks best dive and snorkelling spots of day according to weather and tide. Good value too, considering you get one night's accommodation and two full days on the reef.

Cairns city does not have a beach, but it does have a well landscaped manmade stinger and croc free lagoon, north of the city are beaches such as Palm Cove, Trinity Beach, Kewarra, Yorkey's - all with various types of tourist accommodation. Just north of Palm Cove is Ellis Beach (you'd need a car, its just off the suburban bus network) with beachfront family bungalows - www.ellisbeachbungalows.com.au