We don't have a good solution for the "people who never become adults", especially at the levels where they generate a lot of negative externalities, but institutionalization or prison seems excessive.
When a person isn't of a sound mind, they are still expected to maintain their responsibility but they may simply be unwell. Unable to be responsible for themselves or their actions.
We have various ways of dealing with this. Contracts are not enforceable (or can be deemed unenforceable) against people who were not of sound mind when entering into them, meaning if you are contracting with someone, you have an incentive to make sure they are of sound mind at the time. There are a bunch of bars you have to clear in terms of mental capacity and non-coercion and in some cases having obtained independent legal advice, in order to be able to enter into a valid contract, depending on the context. As well, if someone is unwell but self-aware enough to know something is wrong, they have the option of granting someone else power of attorney, if they can demonstrate enough capacity to make that decision. If someone is unwell but not aware of the extent of their incapacity, it is possible for a relative to obtain power of attorney or guardianship. If a medical professional deems someone not to have capacity but there is no one able or willing to take on decision making authority for that person, many polities have something equivalent to a Public Trustee which is a government agency empowered to manage affairs on their behalf. If you show up in hospital and you are not well enough to communicate your preferences and/or a medical professional believes you are not of sound mind, they may make medical decisions based on what they believe to be in your best interest.
These may not be perfect solutions (far from it) but "you're drunk" or "you're a minor" are not the only times when society understands that people may not have capacity to make their own decisions, and puts in place measures to get good decisions made on behalf of those who lack capacity.
Of course (and possibly, correctly) many of these measures require clear demonstration of a fairly extreme/severe lack of capacity, before they can be used, so there's a gap for those who are adult in age but childish in outlook.
I think this could benefit by expanding "responsibility" into more specific dimensions of acting and decision-making. Nobody else can be control, in the end, for what happens in one's head, or what words come out of one's mouth. And it's meaningless to say that someone is responsible for something they only have a light influence over.
If you break it down into "provide money and other resources" and a range from "advise and guide" to "incarcerate and physically restrain, possibly with explicit punishments for some choices", it gets clearer about what you're thinking regarding solutions. Are there other elements of responsibility you're considering?
Honestly, I don't know of any scalable, affordable, morally tolerable solutions for unparented children (because the parents are absent or themselves not capable), or for long-term incapable adults.
I see a modern puzzle of "self responsibility" which gets tricky quickly. At all times a person has responsibility to themselves. From child to adult, you are “self responsible”. It always applies.
When a person isn't of a sound mind, they are still expected to maintain their responsibility but they may simply be unwell. Unable to be responsible for themselves or their actions.
As a society we prepare for a few times when people don’t have responsibility, for example being drunk or a minor. For minors, we expect parents to care for them until they are old enough to be responsible for themselves, we build schooling institutions to guard children for a while too. For drunks, we make duty-of-care (temporarily) as part of the deal for the people who serve alcohol.
But we as a society also are not prepared to deal with people who are not drunk, not underage and also not self responsible. People who might be able to be self responsible but just aren't. Immature or having a mental break, Society isn’t ready for them.
At this moment in time, there's no one else who automatically assumes personal responsibility for you. There's no service or system that is designed to stand in for self responsibility, nor is it something we can expect to systematise permanently or somehow magically assign to a service.
We can implement some solutions. We can set up expectations. We can plan to support people to live a neutral/positive existence, force them away from negative existences (by outlawing drugs, gambling, violence etc). But ultimately pushing people into neutral/positive existences (aside from being an unhealthy attitude of control over people's autonomy to make bad choices) would be leveraging goodwill from the humans whose job it is to provide services like medical or psychological counselling or other support services. If there exists in the world, a black hole of self responsibility, it's very hard to fill for an individual and especially for a society to do it on behalf of an individual.
Unless someone else comes along and says, "I'll do responsibility for you for a time, until you can take it on". But people who do that are likely to want to benefit from their efforts.
This is independent of the specific condition that may cause a lack of self responsibility.
Beyond basic responsibility, a more mature adult can take more responsibility. They can extend their capacity further and further, leading empires and communities and visions that extend far out. The scary truth is, some people never become mature adults. Some people dip in their mature capacity for a moment, and other people climb up and then keep climbing…
Maybe the most important kind of positive expansion of self responsibility is when one person expands their responsible capacity to have enough responsibility to cover for a partner, and beyond that, potentially to children as well.
It’s possible (biologically) to have a child before having enough space capacity for personal responsibility to be able to do good by yourself, the child and the rest of society, but ideally there is enough space, time, money and all the other resources needed to have personal responsibility for yourself, your family, your friends and your children.