There's always a large crowd that gathers on the pier-side of the ship to watch for pier runners.
On one cruise (and I can't remember the port, but it was a long pier), there was a guy trying to get his extremely-inebriated female companion back to the ship. He was a skinny little guy and she was considerably larger than he was.
He was tugging away at her and she'd decide she wanted to stop and look over the side of the pier. Then she wanted to sit down. Then she wanted to go back for another drink.
The deck hands were beginning to rattle away at disconnecting the gangway and he was getting more and more panicky. She just wasn't ready to get back on the ship. People were yelling, "Leave her! Save yourself!", which, of course, wasn't helping the situation.
Finally, some kind ship's employee rolled out a wheelchair and they managed to heave her into it and onto the ship. Much cheering from the rail.
The other funny time was when we were docked at Cozumel. Directly across from us was an almost-identical Carnival ship, just with a different name. The other ship pulled away and we were due to leave in about an hour. We were hanging out at the bottom of the gangway and a youngish guy and a much older woman showed up to board. Unfortunately, they had been on the OTHER ship. They were both pretty wasted and didn't seem to be particularly upset about it. They wandered off, probably back to whatever bar they had overstayed at.