In Miami the ships board on the starboard side but may dock in various ports on either port (left) or starboard (right), depending on their pier assignment. This often happens based on priority, fee paid or on a first come first served basis
In other embarkation ports the ships board on the port side.
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The logical answer to the original question is - the side next to the pier.
But the answer is that you never really know. I assume a captain might have an opinion about this, and on certain itineraries one side might have an advantage, but the truth is it is like asking which direction your car usally faces when you park it in a parking lot - it depends on which space you choose.
Ships do not use the same locations on ports, sometimes another ship beats them to their spot. But for an NCL ship in Hawaii, you might find the Port side being used more than the other.
I am the editor, but I also speculate, ask questions and play devil's advocate. I reserve the right to change my mind.
I am not sure if you get more on one side than the other, I think it just depends on where the ship docks. I am not sure if there has ever been a survey done on the topic but I will surely post if I ever find one.
When I booked the Carnival Glory a couple of weeks ago, I asked this same question of the Carnival PVP. I was booking cabin 2465 or 2462 (pix window) and I wanted to which side was going to be facing the dock most often. He told me that there was no way to know. It all depended upon which slip the ship was assigned when it was ready to pull into each port.
I would think that the captain would try to have it evenly divided so that each side's balconies and windows got to enjoy looking out to the island as much as possible. However the final decision lies with the port's authority when assigning berths for the day.
(cruzin' solo by design)
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