I second the advice about going early or late. That said, if you have to go midday, keep an eye out for kids and objects that might roll into your path, maintenance workers, etc. If you don't get too "zoned out" you'll be OK. Also, if you're coming up behind someone and want to pass and they don't know you're there, usually a polite "excuse me" or "behind you" as you're coming up on them will do the trick of getting them to slide over a bit.
One other warning: don't run on rough days at sea. In the bow of the ship is a sloped ramp. I went running on a rough day and I was down in the bow of the ship on the ramp when we must have gone over a wave. All of a sudden, the ramp just drops away and then just as suddenly it came back up right as my foot was heading down. I got a nicely jammed knee that was sore for a couple of days. So be careful!!!