I'm going on a luxury yacht charter to Turkey, Greece and Croatia in the spring. Do I need to get any permits to sail in these areas and, if I do, is it my responsibility or the captains to get them?
It entirely depndes on the yacht charter you book.
If it is a bareboat charter (where you will be sailing the boat yourself) you will need to bring with you all relevant permits and licenses to prove that you are qualified and capable of sailing the boat yourself.
Anything that has to do with the legal authorisation to cruise in a given country, and its territorial waters, is bound to be the responsibility of the Captain, as long as it’s all about the vessel and her legal status — whether there is a need to operate under a license or not.
Charter yachts that are legal to charter in Turkey, Greece and Croatia will have all the necessary documentation onboard, and your broker will confirm with the Captain prior to the charter whether there is any potential issue there. This is why it is important to establish your itinerary before departure : do mention that you want to go to Greece, if that is the case, as you could end up realising once the charter has started that the yacht you selected for your charter cannot legally cruise there ; and by then it would be too late !
Please also note that what applies to the superyacht may apply to you too : you maybe required to hold a valid visa to enter certain countries, it is always best to check beforehand whether you have complied with your own legal duties prior to embarkation. It would be a shame to miss on some amazing cruising opportunities and change plans simply becaue you forgot to apply for a visa.