Locals from Korcula usually visit Badija by their boats while visitors to Korcula and surrounding area are usually taken there by local taxi boats.
Taxi boats are all lined up in Korcula Town, in Punta Jurana area, at the Korcula’s eastern pier, so if you are after visiting Badija, this should be your starting point.
In terms of accommodation, although Badija used to have its own hostel, that used to be located in monastery building, unfortunately, there is currently no any accommodation on offer on Badija, which hopefully might change in the future.
Instead, if you would like to enjoy daily views of Badija, you would need to stay in the surrounding area.
Soline offers great views over Badija Island (see the location of Soline) as well as some really nice accommodation, like apartments and rooms at Villa Sole. Villa Sole also offers small boat hire service for its guests, to enable them to sail around the archipelago.