This is kind of a tough question, just based on the geography of the country. Dubrovnik is situation in one very tiny corner of Croatia, so it is certainly not centrally located at all to be a "home base" (for day trips, for example). The country really doesn't lend itself to doing a lot of day trips from one hub (and the lack of a car wouldn't lend itself to this, either).

In my opinion, this place is best seen by moving through it - not really having a "home base" but starting at one end, and moving up or down the coast by either ferry or bus or car or whatever.

Now, if your question is driven by the fact that you will be spending an extended period of time in Croatia and you are looking for the best place to set up a household, I don't know that I would recommend either Dubrovnik or Split. But that's a different discussion.