Bournemouth West is the closest section
Or a little bit further is Durley Chine
bournemouthbeachhuts.co.uk/durley-chine.php (click on the arrow at bottom to see more of them)
If they are using a car to get down there then Durley may be better so they can park. If happy to walk and use Zigzags or lift then either will do.
The closer you get to the pier the busier it will be is the general rule of thumb. Also the prom can be wider in certain areas so if you are trying to keep an eye on kids and avoid cyclists at certain times different areas all have their own benefits. The further you head from pier the quieter it will be.
You get a key yes but depends how brave you are as to what you leave in there, it's only a padlock and some windows between your possessions. people do leave a fair bit down there and crime is relatively low but don't be complacent. Also bear in mind if you hire from the council you will only have deckchairs and a cooker in the hut, everything else will probably have to be carried down there(or driven to the hut outside of hours, i.e. if you want acup of tea you need your own kettle, cups and spoons)). An alternative is to rent a private hut where generally they will have furnished it with the basics, in some cases down to having kids toys/spades etc. Often the prices are better aswell plus you can speak to an owner about it and get an idea of what it is like.
Search for "Durley" "Bournemouth Pier" and "West Cliff" in the below site and you should get a few selections