If you're after old-fashioned romance ... near the port. And if you simply want to enjoy quieter beaches and maybe make a day trip into Dubrovnik, you'd be best placed in Mlini or Srebreno Bay.
If you're arriving to Dubrovnik via cruise ship, you can take a taxi from the port to the Old City for about 70 kuna (around $10.50) or reach the area by bus. The main bus station is located at ...