We have not been to a nude beach but at Valley View HS. We leave our cloths in the car. I carry a back pack for my SLR and I put our valuables and keys in and lock the car. I have started carrying a tactical knife in the pack as well.
Into my backpack. Im never worried about someone taking things, as whomever is near us on the beach will know who belongs to the items there. We usually have a sun shelter that everything gets tossed into the back off, and most thieves aren't going in there because others will know it's not the people who everyone took notice of, setting the thing up.
We usually leave them in the car! And sometimes we leave them at home when we are feeling daring! We have also left them in the meadow, along the trail, hanging on bushes. Sometimes it can be a bit of challenge to find them later though, More than once we had to drive home naked. Which is a different thrill than planning to drive naked.
My wife does not sea swim so if she is with me I have no problem as she will look after my clothes.
The only times I have gone to the beach alone have been on holiday and in that case I have always left all my clothing in the apartment or hotel room in the care of my wife.
Edited to add
I have a video of walking down for a swim here.
We have a backpack with our towels and sunscreen, so the shorts, t-shirt, and flip flops go in the backpack when the towels come out. Our car has a push to start ignition, so the "key" is just a small fob, which goes in the pocket of my hat along with a couple of $20s. Everything else stays in the trunk of the car. This system works for the beach and local camps. At Cypress Cove we get a wrist fob to unlock our room where everything but the towels and flip flops can stay. On nude cruises I wear my towel on the tender from the ship to the beach when we go to one of the private islands. Then I only need my ship to shore card and picture ID, which go in the same hat pocket. Plus, who would steal our stuff when everyone on the beach is from the same ship?
The last time we were on Orient Beach we took a taxi from our cruise ship so we could see first hand the destruction from hurricane Irma. That day we had more clothes plus our wallets, so we picked two beach chairs near the guy renting them so somebody would be watching our stuff while we were walking around. Threw our clothes in the backpack and then threw our towels over the backpack when we left the chairs. In the past we would have rooms at Club Orient, so wallets and clothes would stay in our Mini Suite and we would charge everything to our room, so a key (with a coil wristband) and a towel would be all we would have out on the beach.