Or if they just don’t. For whatever reason. Stick with “What do you do?”.
I like “where are you from?”
If the answer is not here, I’ll follow up with “what brought you here?”
This gives you a variety of things to talk about, and you’re being genuinely interested in the person you’re talking to.
“How do you know the host”, “Do you live near here”, and “Are you from around here?” are my usual go-tos.
I would definitely suck up to the plumber, much more inportant!
Ask: “What’s the worst thing you’ve ever done in your life?” They’ll be so shocked and uncomfortable, they’ll be glad to tell you what kind of work they do.
When I was in college, 50 years ago, the number one question at parties was “what’s your major,” in other words what are you going to do.
I’ve lived in the Northeast for most of my adult life and have never been asked this question myself. However, according to my officemate who grew up in Georgia, it’s a common icebreaker in the Bible Belt, the assumption being that most people are churchgoers. She says that for Southerners it’s like asking people where they live or where they come from (because a church is like a home) or what they do, with some churches (like Episcopal and Methodist) being seen as higher class than others which are perceived as more working class (like Baptist)
I agree with ZonexandScout and rbroome that “How long have you lived here?” is a relatively neutral starter.
And if the person says ‘I don’t have children’. What next?
I think there’s plenty of reasons people may not have children and may feel uncomfortable about the question - they’re hardly going to say as much to your face.
But the only reason a cocktail party happens is because everyone there is a classist. 
Any occasion where people who don’t know each other are standing around aimlessly and pretending to sip alcohol , the only thing on everyone’s mind is “hmm…who can I out-class now”?
That’s what a cocktail party is.
And the few that I’ve ever been to have been insufferable. (Openings of art exhibitions are the worst.)
Other occasions can be better and less awkward. Say, a co-worker’s wedding reception, where you don’t know anybody. But at least you know the bride’s name, and you can open a conversation with a stranger by asking them how they are related to the bride, or something similar.
In the alternative, the person says, “I don’t have a job.” Same problem. Or, maybe the person’s job is to fire people all day long. Or the person’s job is to kill pigs all day long.
There’s no perfect question. There are only questions that are more likely to be successful. For example, if I knew or suspected that a person is homosexual, I would not use the children question, because homosexual people are less likely to have a child.
100%, hence the question posed by the OP.
Of course there’s no perfect question, but there are some some subjects which can make people feel genuinely uncomfortable or even upset - jobs and kids being two such hot potatoes. ‘Where do you live, how long have you lived there, what’s it like’ etc etc, are on much safer ground. Also, I tend to wait for people to drop clues - if they mention their kids, or taking their dog for a walk, or working for the local council, or their recent trip to Europe, or knitting, they’ve given me a way in. People find it hard not to mention at some point the stuff that’s important to them.
(Of course this does mean you can get stuck with someone who talks for two hours about his folk singing group - yes this did happen to me recently.)
“What movies have you seen lately?”, because most people like movies.
The definition of a good conversationalist is someone who can get someone else to talk. Once I was at a party and I noticed someone else who was also wearing a tie with Loony Tunes characters on it - Bugs Bunny and so forth. I pointed to it and said “Let me guess - Father’s Day” because that’s where I got my Loony Tunes tie, and it led to a twenty minute conversation where everybody was talking about the stuff they got for Father’s Day/Christmas/birthdays.
Regards,
Shodan
I like to ask where people live and how long they’ve lived here/there. I like hearing about other local places, or why people came here to Cleveland. And if I’m at a place where the person is not likely to live in Cleveland, even better.
I actually really like talking about other peoples jobs, because they fascinate me. Like, every job fascinates me. Heck, a couple months ago I talked to a guy who came from Costa Rica to live in Cleveland to dig big hecking tunnels for sewage. Totally fascinating conversation!!
Personally I don’t like to be asked what I do in my spare time or my hobbies, because I don’t do anything. I mean I watch TV. I don’t want to talk about that.
“Have you lived in [the city we’re in] all your life?”
I also really enjoy learning about other people’s jobs, and I don’t think I’m being classist by asking, but I imagine my ingrained biased are at work at some level.
My “do you live nearby?” ice breaker is probably more classist than asking about the person’s job. The neighborhood you live in probably reveals more about your class than what you do for a living.
Go ahead and try asking kids what they do. You get way better answers from them than you do from their parents.
I don’t think there is a perfect question, but I do think there are questions that aren’t likely going to make someone feel some kind of way. "Do you have children? has a good chance of doing that for the aforementioned reasons. But “How long have you lived here?” likely isn’t.
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I like “Read any good books lately?” because I love to read and can get great suggestions that way. It’s probably the most frequent conversation starter among my family members. It also can lead to really fascinating conversations.
Not yet.
This subject feels rather adjacent to the “cold reading” that psychics do.