I believe someone mentioned Portsmouth, would that be a better fit for a family? The purpose of visiting the coast is simply to see the coast. Days spent on the beach is not the ultimate goal. What would you suggest based on the short time we plan spending on the southern coast?
That was me. I live in Portsmouth so my outlook can either be biased or jaded depending on the advice I'm giving!
Portsmouth can be reached by direct train from Gatwick airport. If your only purpose in visiting the coast is simply to see it then Portsmouth would be fine. The main draw for tourists is the Historic Dockyard which is a fantastic historic attraction and there are activities for young children, primarily Action Stations and the rigging climbing, I suspect the children may find touring old ships a bit boring but you know your children, perhaps the 4 year old would love running along the decks of HMS Victory.
A guided boat tour of the harbour is also recommended and will allow you the opportunity to see many of the naval ships that are docked.
You can walk the length of the seafront on the esplanade and you'll find Clarence Pier, a funfair and amusement arcade which has passed its sell by date but I'm sure will still excite your children (think Atlantic City's Steel Pier rather than Busch Gardens). Further along you'll pass the hovercraft station where, with a bit of luck, you'll be able to watch the hovercraft arriving or departing on the beach.
Further along you have the BlueReef Aquarium, a small but enjoyable aquarium, good for young children. Whilst a short walk from there will take you to Castle Fields and Southsea Castle, built for King Henry VIII and from where he watched the battle during which The Mary Rose sank. It's a great little castle for kids to run around and explore and offers great views of The Solent.
From there you can carry on the walk along the seafront to South Parade Pier and Canoe Lake opposite. The pier has a small amusement arcade and during the summer hosts a small traditional funfair. The lake is small, built in the late 1800's and durnig the summer you can rent pedal boats and go zorbing (floating in a giant plastic ball). There's also a splash park for young children and an adventure playground. Next to this are the rose gardens and miniature village, a bit twee but young children love it. Behind the lake is Cumberland House, a small natural history museum with a butterfly house which, in summer, will be full with butterflies that you wander amongst.
From here there are tennis courts and a miniature golf course (not really for children, just a very small golf course). The remainder of the esplanade takes you to Eastney where the beach is left to its more natural state with wild plants and a very discrete nudist beach. To carry on further merely takes you to the residential parts of Eastney, Milton etc. There is a ferry from Eastney that takes you the short distance to Hayling Island. Here the beaches are primarily sand and when the tide is low there are huge sandflats that are great for kids to run around on, interspersed with pools that the tide left behind. It's a very popular area for dogwalkers and horseriders (although horses have to be off the beach by 10 am) and in the summer can be quite busy.
Gunwharf Quays is a relatively new retail and entertainment development next to Portsmouth Harbour Station. It consists of outlet stores, restaurants, a cinema, luxury housing and the Spinnaker Tower, a large tower that was intended to mark the millenium (it was originally going to be called The Milleniunm Tower however in true British fashion it was finished too late and over budget and is now called The Spinnaker Tower). It has a viewing platform with a glass floor near the top that provides fantastic views of the area for those not afraid of heights. There are also a variety of boat tours available.