Bournemouth is under 2 hours by train from London. In the summer, you can get cheap day return tickets from the local train company, or they do group tickets, network railcard discounts etc, so it isn't too expensive to head down outside of peak times. See this answer for more on getting there.
The local council rents out beach huts by the day or by the week. If you go for the East Cliff ones, it's a 10 minute walk from the station. Otherwise, plenty of buses. You have about 10 miles of lovely sandy beach to choose from!
Poole council also rent their beach huts, and are on the same bay. However, Poole station is further from the beach, so you would need to get a bus. However, there are normally open top buses on that route, which give you great views of the harbour and Brownsea Island on the way to the beach, so it's not the end of the world to go there!
There are places nearer to London than Bournemouth with beach huts, but generally those have pebble or stone beaches. If you want a Beach Hut and Sand closest to London, the Bournemouth/Poole/Studland bay is your best bet!