Be aware there is also a town called Supetar but it's miles away from the island which is close to the small town of Cavtat, about 10 miles south of Dubrovnik. There's a regular bus service from Dubrovnik to Cavtat (pronounced "Tzavtat").
It can be reached by boat (obviously!) from the port at Cavtat at a cost of HKR 50, about £7.50 or boat from Dubrovnik for about HKR 100, about £13.
Nearby is the village of Plat (on the mainland) but don't go there. The derelict hotel is dismal and the beach not worth the bother.
Now then. Whether this is the naturist beach or not, I don't know. Suffice to say I was the only person there! It's a tad sandy (as you can see). Getting here is simple. Get off the boat from Cavtat, turn right at the little restaurant - there's nothing else on the island really - and follow the grassy path for about half a mile. Bingo! Be sure to get the last boat back! In the evening it's a disco-party place - I assume at week-ends during the summer.
Lokrum island, near Dubrovnik.
Boat from Cavtat or Dubrovnik (much closer), cost HKR100 (about £13) but my ticket included admission to the botanical gardens. To get to the signposted naturist beach, just follow the path left from the docking point. The "beach" is just slabs of granite. Mmm, comfy! Showers on the beach but that's all - apart from a sign that announces "No Shampoo!". There's a café-type place at the harbour.
Valovine Bay, near Pula
OK! Here we go with the first addition for 2019. The bay is known as Valovine, it's not "official" but sufficiently remote not to be a problem. It's close to Pula airport which is a 30KR (£3.60, US $4) shuttle bus ride away. They run from about 0700 to around 23:00.
The easiest way to get there is the number 1 bus from Pula bus station to Stoja. From here (it's the end of the line, right by the sea) turn right (sea on your left) and follow the track through the trees which follows the bay. Either that or a 45 minute walk from town. Follow the main road and turn left at the City Max shopping mall (sea on your left). Follow the main road. By the bus stop is a fast food outlet, ice creams, a bakery. Parking is free but note - I didn't find any toilets! For the first place, the rock, it's easy to find but a bit of a scramble. The cove is tricky and best not for the less mobile and is a shingle beach; four people would fill it! Don't expect sand in Croatia, there isn't much. Near the town of Rovinj is the resort, Valalta, you may need an INF card if you visit. I didn't go.