I got this topic last week when I did the call for stories, and I've been trying to figure out how best to address it. The short answer is: none outside the photo op. Your photo is the only guaranteed New Kids encounter while you're on the boat. It's just enough time to say hi to your favorite Kid and tell him you're having a good time.
Other than that, I guess it's half timing and the other half's luck. You likely will run into at least one NK over the course of the weekend -- in the elevator, near the pool, walking down the hall -- but that doesn't mean they'll be able to stop and chat. When Joe walked past several of us Saturday afternoon, he said he couldn't stop because he was on his way to an interview. Earlier that day, though, girls ran into him next to the pool and got photos.
The other factor is how bad you want it. Each night (and probably during the day), there was a small contingent of girls that followed the guys around like puppies. I wouldn't recommend that.
I saw everyone except Jordan at some point over the weekend, and I did not talk to anyone. If you are more forward than me, I'm sure you could have progressed to some form of conversation. I tended to act the same way passing a NK as I would passing our room porter: just said "hi" and kept walking.
Of course at the other end of the spectrum was the Katie we all saw in the DVD Easter Egg. Which makes the final answer "somewhere between saying hi in the hall and making out in the casino."
2009 cruisers: please post your face time stories!