I've stayed in Dubrovnik twice. On the first visit, I didn't have a car (actually, I dropped it as I arrived). I stayed at a little soba (a Croatian word for BnB, basically) close to the Pile Gate of the old town (inside the walls). If you look on a map, the Hilton Imperial hotel is close to where I was staying (I stayed across the street in the little neighborhood there). It was a very easy walk into town from there - and no steps, which can be a challenge in some parts of Dubrovnik. Keep in mind that in the largely car-free old town, many pedestrian streets are up an incline with steps - if you stay in the old town or above it, make sure you are clear exactly what kind of terrain you have to navigate. If you are OK with a lot of steps up/down to get to your front door, then it's less of a worry but it might be for some.
On my second visit, when I had a car, I stayed away from the center in nearby Lapad, which is not an exciting area, but it has some restaurants, and there are some public beaches nearby. This was only a short visit; I was able to park my car at the cheap hotel where I stayed and take the bus into the old town (it ran fairly late). Parking a car near Pile Gate or the old town would be in an expensive parking garage most likely.
I definitely preferred not having a car on the first visit, so if you are starting or ending a trip in Dubrovnik, you probably won't need one. Staying in Lapad was practical and worked fine for that trip but wasn't as desirable.
Some people stay in the nearby town of Cavtat which I hear is charming but I haven't visited. Apparently bus service into the old town of Dubrovnik from there is good too. Dubrovnik itself gets mobbed with tour groups during the days but is much more peaceful and quiet at night (DEFINITELY visit the old town at dusk and night at least once - it's truly magical in there when it's all lit up at night, and at dusk the swallows fly over your head seeking bugs.). Try to watch at least one sunset from the Buza Bar, which is in the town walls.