Let's be fair: money can't solve all your problems, but it can solve some of them.
The only problem is that serious mental illness is one of those things that money can't magically fix.
E.G. Does someone like me or do they just like my money? What should I do with my life now that I'm unfulfilled by indulging my whims?
But of course as others have said, it brings another set of worries.
Still, money is an enabler that creates options.
I believe I know what you mean. I disagree. I have no experience being a billionaire, or a millionaire for that matter, however here are my casual observations and thoughts:
A billionaire cannot just quit a job. There must be a transition period. A billionaire can't just no call no show. It's possible, but it never happens.
A billionaire cannot travel to Europe like I can, going from Airbnb to airbnb, hopping in an Uber, surfing with the locals, enjoying fish tacos. A billionaire ends up being an ambassador with little privacy.
Immediately get appointments with great healthcare providers. Maybe. But so can I.
Throwing around alprazolam/clonazepam as a solution for anxiety is trite and can be damaging. Take caution with them.
None of this serves to answer the question of how they deal with it, but rather to confirm what we probably all know: while money makes certain things in life easier, it is not a cure for all ills.
[1] https://www.theatlantic.com/magazine/archive/2011/04/secret-...
Not to be confused with the (damaging) ideas of the "prosperity gospel", Andrew Carnegie wrote https://en.wikipedia.org/wiki/The_Gospel_of_Wealth , and now we have libraries.
Bill Gates is busy trying his hardest to stomp out malaria.
Paul Allen is trying his best to do as much blue-sky biomedical research as possible.
etc.,etc.
Even if, perhaps especially if,you came into your riches by unscrupulous ways, seeing that your surplus wealth is sunk into things that matter will help you sleep better at night -- because, indeed, you're doing what you ought to be doing.
For the most part they don't have the day to day anxieties about money and what things cost of course. And also they have less of the sort of existential dread that their lives won't matter, as they are surrounded by a lot of evidence they are important.
From what I can tell the main anxiety comes from the drastic effect it has on interpersonal relationships. When you have a tremendous amount of money and status you necessarily have a very hard time trusting people, and you're likely to face a near constant inundation of people who want things from you, and who often are being disingenuous about their intentions.
It's exhausting. For the most part they deal with those anxieties by finding ways to get out of that situation by surrounding themselves with people that have similar money and status, removing themselves to isolated or exclusive places, and so on.
Here's one of the studies: https://www.ncbi.nlm.nih.gov/pubmed/16542786
But, there's several more.
Back to the question: With all their money, they could probably afford to eat the very best foods and could higher a nutritionist to help them plan high magnesium foods. For everyone else without all that money, there's always my nutrient based food/recipe finder: https://kale.world/c
Use it to find the most Magnesium dense foods or any other nutrient for that matter.
So I will suggest that in some cases the answer may be that becoming a billionaire is part of how they dealt with their anxieties.
The classic "mo money, mo problems" sounds (and is) ridiculous to people without much money, but after a certain threshold it actually becomes true.
I think that billionaires have better ways of dealing with it than poor people.
Money can certainly bring you suspicion of others and that can make it as hard as it does for the rest of us to get things done!
It also doesn't buy self-awareness, one of the most important starting points out of anxiety.
Take care of your physical health, and your well-being -- including especially your day-to-day environment and social relationships.
Money makes some of this easier. On the other hand, if you don't do these things, no amount of money will fix it. (And many health problems, no amount of money can fix or undo. You want to prevent them occurring, in the first place.)
P.S. I do know a few significantly rich people who seem fairly happy. And the above is what I notice.
Financial constraints -- and bad teaching and communication about same -- kept me from addressing and moving away from some very significant stressors. Anxiety became severe and chronic and fed into a downward spiral. Despite having very high intellectual marks, from multiple sources, as well as the reputation for being a nice guy (hopefully, without the quotes: "nice guy").
Actually, earlier than that, family and school compelled me to endure chronic abuse and bullying, with no effective help from the responsible adults. That seemed to initiate the core of the anxiety.
Both the family and the rather well-endowed school system plausibly had the resources to make things different, but they didn't.
Back to my point: Pay attention, and use the money to take care of yourself and your environment. This also makes essential social connection easier -- feeling better about yourself. Best proposal I have.
There are no-money-problems and to-much-money-problems. I think the later is often misunderstood. Lawsuits are usually a pain, not only for billionaires but for wealthy entities in general. If people know you have a lot of money then they'll try to get a piece of the pie, including the government. So yeah, might be a little stressful.
Though my guess is that if you made to the top and became a billionaire from your hard work, then you can certainly handle stress and anxiety. You might even be a freakin guru at it.
+ Diverse investments
Instead of keeping all their eggs in one basket, they tend to be invested heavily across multiple spectrums. Oil and gas, health insurance marketplaces, real estate, etc. Even if one facet of their portfolio crashed, they wouldn't be screwed.
Many of them will even make well known subpar choices to add some "insurance", like keeping a safe full of metal silver and gold bars, or even investing in a well supplied safe room. It's not that they're expecting to use those things or make a profit from holding rare metals, but to them $20k is nothing to invest in a "just in case" fashion.
+ Multiple experts
You or me might have one doctor, they might have three or four they get opinions from. Likewise for accountants or lawyers. This gives them flexibility, better chances for good outcomes, and less likelihood of a single person being able to screw them over.
+ Outsourcing to experts
Most of us probably only go to a doctor or a lawyer when things get way out of our depth. Likewise for an electrician. Someone who is wealthy can afford to apply experts to every problem they have. Instead of only going to a lawyer when they really think things have gone wrong, every single contract can be reviewed by an expert.
+ Concierge services for when things go wrong
Ever have a flight get massively delayed or end up in the wrong city? For most of us, it sucks and we scramble to deal with it. The wealthy can afford custom concierge and assistants who can fix things in a single call. Flight stopped in the wrong city on your way to your exotic adventure? No problem, one call and you'll have train tickets and a cab booked so you don't miss anything.
This applies beyond travel. Health crises in a foreign country? Call the hotline and have a doctor show up at your hotel instantly. Lost in the mountains while hiking? Pick up the sat phone and have a rescue team scramble to your location.
+ Amazing amounts of connections
When we go to a museum and there's a crazy line, we have the choice of waiting or walking away. Someone who is wealthy and a savvy donor can call and arrange a private tour instead. Likewise for getting into sold out shows, better seats at plays, or even booking tables at restaurants. For someone massively wealthy, donating a few thousand a year to local art institutes not only looks good and has tax benefits, but costs relatively nothing and comes with amazing benefits.
Likewise, they get to mingle with actors, celebrities, important leaders, etc. Suddenly the rolodex goes from a few well known friends to an impressive array of people with amazing amounts of power.
There's a big concern that your money might be used illicitly or illegally, or even if you try to give it to a good cause, that it may be squandered.
You also have to worry about nutjobs who find out you have money - this leads to security, both guards and changes in how you live your life. If you have kids, their lives are probably very sheltered since kidnapping is a very real and very serious threat. Likewise for close family members.
The value of money is always a diminishing return, and eventually it can become a liability.
Less snarky: They have different anxieties. They don't worry about losing their house or their job. They don't worry about the monthly bills or an unexpected medical event. Instead, they worry about being back in the place where they have to worry about those things. (I think. I'm not, alas, in a position to know first-hand what billionaires are anxious about.)
So they spread their money around. "Divide your fortune into seven, or even into eight, for many are the mischances of the world" is good advice. (It's from Proverbs, in the Bible.) Billionaires do this, so that they can comfortably survive any particular investment going horribly bad.
Then, I suspect, the more aware billionaires worry about their effect on people. They can throw money at friends' and relatives' problems, but that may actually cause more harm than good. I don't know how they deal with that - maybe by having a fixed policy of what they will and won't do, and not deviating from it no matter how good the story sounds.
Without that purpose to pursue, a billionaire might feel anxious about what to do with their life.