By car, you can drive to Elafonisi all year round.
Bus trips, as well as boat transfers or cruises are only available from   May to the end of September/October. You won’t be able to go to Elafonissi in winter if you don’t have a car.
For me the best time to visit Elafonissi is May, June, September and October. You can swim, enjoy the beach while avoiding (a bit of) the tourist crowds of July and August.
For a visit to Elafonissi in summer, expect crowds. Go there in the early morning or at the end of the day if you want less of a crowd. Elafonissi beach is also superb at sunset.