Good responses. My experience will be similar to BR2011.
My first tour was on a 210' cutter. While it's a 210' medium endurance cutter, which would be small in most services, it's a decent size for the Coast Guard. We had a wardroom (officers) of 12 and an enlisted crew of 63.
I was the assistant operations officer, so I managed a few divisions with maybe 12-15 folks, bridge BMs, OSs, ETs and an IT. For my bridge BMs I had a chief and a few mid-level petty officers. For the rest, I had an OS1 (eventually an OSC) and an ET1. We had some good leadership. I made my mistakes Early on I bypassed by OS1, but eventually understood the errors of my ways. Our divisions were close, made even closer by a relatively difficult operations officer (my boss) we worked for.
In their spaces the relationship was "friendly." As in "IT2 how's it going..." "Good Mr. LITS."
I didn't have a first name relationship with my guys and I wouldn't have thought it, in that setting appropriate. I used their ratings, like BM3 or ET2. That doesn't mean they were rolling out "sir sandwiches" but generally there was a sir in there every now and then (as I used for my boss, the XO and the CO).
After two years on a ship, in a fairly organized structure I went off to Coast Guard Headquarters where half of my co-workers were civilians, 1/4 were officers and 1/4 were enlisted. This was more of an office setting, typical of most agency HQs in DC. I still referred to my enlisted co-workers as "Master Chief" or "PA1", and I was generally "sir". Civilians were first names. Some referred to the enlisted by their ranks, but usually just names.
In the military, specifically the Coast Guard, for me, the situation matters. As BR2011 said, the size of the unit matters. While a seaman on a 87' cutter has a closer relationship with his commanding officer, the seaman on a 378' likely rarely interacts with his captain. A small land unit, or an isolated unit, is closer than a large unit or a unit in a highly populated area.
Even between units the relationship can vary, but the general relationships remain. It has to be professional. It has to be work-related. You should not socialize or hang out in your private life.
As we pulled into ports, especially small ports, the officers went out with officers and the crew went out with crew. Sometimes the groups would run into each other or end up at the same bar, but it generally wasn't (nor shouldn't have been) one going out with a group of the others. Don't ignore them, by a "hey!" or "how are you guys doing tonight" is fine.
And don't think they really want to. They work with you and there are times they don't want an officer around. They don't want to chill with their boss during liberty in a port. Don't feel bad about it. There are things that come up between officers, including disagreements or issues, that don't need to be the gossip of your enlisted shipmates.
After saying all of that, you watch out for your shipmates. If you see a junior enlisted shipmate drinking far too much and starting to get in trouble, don't be affraid to take him under your wing and back to the boat. Or, tell one of his buddies to remove him from the situation. You're responsible for each other, officer and enlisted alike.
Relationships are interesting though. For the Coast Guard it isn't just an "officer v. enlisted," but also variations on the relationships between different divisions or departments, the crew and the chief's mess, the chief's mess and the wardroom, the wardroom and the crew, and even members of the wardroom. I might call my buddy "Bill" in our stateroom, but on watch and when referring to him with a member of the crew, even though we went to college together and were the same rank, it's "Mr. Johnson."
But as I said in my first post, these relationships, and the tone between the officer and enlisted ranks varies between the Army, Marine Corps, Navy, Air Force and Coast Guard.
), there is a temptation to be social with some of the same-age enlisted that work around you. Sometimes you can be sociable, but you can't be overly friendly (e.g. if someone thinks there is favoritism or such, you went way too far). I've had guys invite me to go drinking with them on the weekend, and politely declined. There have been times where I was at a training class, and the class (just a few people) had a cook-out and traded stories over beer late into the night. I've also been part of a big group where there were mixed O's and E's out on the town. That's a bit grayer, and you need to be considerate of potential issues. That time, I enjoyed the night, but made sure I never got too drunk. I've seen people do stupid things after drinking too much! My advice is to think twice anytime you've got non-peer ranks and alcohol. 
