If you want to stay in Split, then you want to stay in the old town, by the coast and the UNESCO palace site. There isn't a better area than the road where you'll find the Peruzovic apartments.
Right by a small plaza with bars, restaurants, convenience stores, and a pharmacy, they're set back far enough to get some respite from the crowds but are still a very short and easy walk (~5 mins) to the scenic Riva waterfront. Mediterranean beauty all around.
As a base to explore Diocletian's Palace (evolved into a gorgeous, bustling district of its own, lively and friendly at day or night) and the huge Marjan forest park, or to catch a catamaran to Hvar, etc. or a big ferry for a longer journey, it is ideal. The beach isn't as close, maybe 15 minutes on foot. You may want to uber that. It's pretty, though.
The apartment was very clean and modern, with good cooling and an impeccable bathroom. The fridge was handy, as were the closet and table, and there was a safe too. No cooking options, but with eating out being so affordable, close, and delicious, it isn't any loss. The beds were entirely comfortable, and I appreciated the kettle and coffees!
There is a nice outdoor seating area at the main building, which is right by the extension. You may have to climb flights of stairs, so do check that if a lack of an elevator may be a problem for you.
Last but not least: this is a family business, and the family is brilliant. Helpful and lovely all around. My big thanks to them all!