I just started a new job and most of my new fellow coworkers are very active on LinkedIn. Is it rude or strange to connect with the ones I've met and worked with so far, or should I wait until I develop closer relationships with them? I'm relatively new to "white-collar" networking of this sort and I don't want to treat it like mere social networking. I certainly don't want to put off supervisors if this kind of behavior would be inappropriate.
Here's the approach I've both used and seen: once you know you're there to stay, go ahead and connect with your immediate coworkers. Then wait; if they're as active on LinkedIn as you say, other coworkers will pick up on this and start sending connection invitations to the new guy. Later (weeks or months), review and add anybody who got missed who you want to be connected to. Most places don't expect you to be connected to everybody anyway, so no need to obsess over that.
When I joined this company I had about a dozen immediate coworkers, another dozen in the next "ring" (worked with sometimes or occasionally), and a couple hundred beyond that. It would have felt and looked weird to find those couple hundred and connect in a big batch; let those grow organically. (I say "looked weird" because LinkedIn publishes connection updates, so your connections will see "250 new connections this week".)
I would wait until your probationary period (3 months usually, but I would wait 6 months just to be sure) ends before you do this. Make sure they trust you and they know you are competent, then begin adding people. In my experience, this has caused no issues with anyone. Additionally, make sure you do this after the work day, so people don't think you're screwing around on social media sites during the day.
After all, they wouldn't be on LinkedIn if they didn't want to network with people. That's what it's for!
Just to weigh in with a totally different opinion, I think it depends on the culture at your company, as my experience is totally different from the other answers above.
Where I work, people tend to add connections on LinkedIn immediately upon meeting someone. Even if it's someone you happened to meet just one time on a training course, even if it's your new boss and you haven't even started the job yet. That just seems to be our corporate culture though. Although I'm UK based, its an large American corporation and our company has a big focus on building your personal network (as in, your general network, not LinkedIn specifically!)
Yes, people do tend to connect just for the sake of connecting, but I do think it can be useful for remembering names/faces, and for finding out what people's career and academic backgrounds are, which I find quite useful. e.g. if you're an software Engineer and you're about to give someone a high level overview of a software project, and then you see on LinkedIn they've got an BSc in Engineering and an MSc in Computer Science you might decide to include more technical details than you otherwise would.
I've had senior leaders contacting me via LinkedIn to have a chat out of hours! Something that wouldnt' be possible by work email and wouldn't be appropriate by Facebook. It's been a good way to find allies. Some senior leaders DO use LinkedIn to check out their employees and if they like what they see, I do think it can create opportunities for you.
But again, it all totally depends on your workplace culture.
Yes, wait until you have closer connections with them. Frankly, LinkenIn is pretty useless as a social network except when you're between jobs, and there's no reason why you shouldn't forget about it until then. I tend to connect with colleagues when I've either just left a place, or if I get close to them as part of my work. Adding people just to have a bigger network is no better than people with thousands of facebook friends.