September in Dubrovnik can still be very warm, please note that it is also much wetter than July and August.
The highest average daily temperatures in September are still a lovely 24°C (about 75°F), and the lowest temperatures are around 15°C (about 56°F).
The tourist crowds are less, the amount of average daily sunlight still being about 9 hours, and the average sea temperature at around 23°C (about 73°F), it is still perfect weather for spending time at the beach and swimming.
Keep in mind that, although the weather is still generally beautiful, the average daily rainfall increases a lot between August and September.
Weather forecast for Dubrovnik and Croatia: http://www.dubrovnik-online.net/english/weather.php
Cultural and other events still take place in Dubrovnik during September, the foremost being the Julian Rachlin and Friends Chamber Music Festival.
More information about this festival at http://www.rachlinandfriends.com/