Kotor is at the head of a long deep bay and the water is not as warm or as clear as the open Adriatic — the bay is fed by mountain rivers and tends to be a few degrees cooler than Budva or Hvar. For real beach swimming most people head 25 km south to Budva.
That said, there are pleasant local swims. Dobrota has a continuous public promenade along its full five kilometres, with frequent steps and ladders into the bay and a couple of pebbled stretches. Muo on the opposite shore has a small beach at its southern end. Stoliv further along is quieter still and the best place for a swim with a view back to the old town.
For sandy beaches in the proper sense, take the 20-minute drive south through the Vrmac tunnel to Tivat or further to Pržno and Sveti Stefan.