You can create your own String representation of the array like this:
StringBuilder sb = new StringBuilder();
for (int i = 0; i < playlists.length; i++) {
sb.append(playlists[i]).append(",");
}
prefsEditor.putString(PLAYLISTS, sb.toString());
Then when you get the String from SharedPreferences simply parse it like this:
String[] playlists = playlist.split(",");
This should do the job.