Questions like these are extremely important to me, both as a Marxist and as someone who has dealt with mental illness as a permanent fixture in his life. It has affected my parents, my siblings, my aunts and uncles, and myself personally, as well as many of my friends and former partners. And it is a question that many, dare I say most, Socialist organisations utterly fail to address correctly.
Let me start with a very plain and very clear matter of fact: Mental illness will not disappear post-capitalism. Many socialists believe that it will, but frankly these socialists lack comprehensive understanding of the machinations of mental illness, and I believe in many ways the position of "it will disappear under socialism" is inherently utopian and idealistic.
A major contributor to most cases of depression and anxiety come from the material conditions of capitalism. The alienation of labour being chief among them. The poverty of the working class and poor being second. One of the largest contributors to suicide rates are economic conditions. Usually in the form of debt. Another major contributor is long-term unemployment (to all of the above). The psychology of a person suffering depression/anxiety as a product of unemployment comes down to labour as a commodity, and therein individual value being rooted entirely on the productivity of their labour, their existence as a labourer.
From the commodification of labour we also see a very straight line from the exploitation of labour to the exclusion of individuals incapable or less capable of productive labour. That is to say that the elderly, physically disabled, mentally disabled, the sick, the terminally ill, unskilled and long-term unemployed become objects of neglect and are often discarded. And it is this reality that makes welfare from the state so necessary.
Post-capitalism much of these problems will cease to exist. And therein the pandemic of depression and anxiety will alleviate. But it will not disappear entirely because the components of these illnesses, and mental illness in general, are not entirely rooted in the relationships of capitalist production, or environmental conditions.
Some of the core components to depression and anxiety are experiential. Loss of loved ones, failed relationships, feelings of personal inadequacy, negative life experience, violence, sexual difficulties, and of course hereditary factors (which are still the object of much research, it is understood that there are hereditary factors, but it is not understood what they are exactly).
Treatment will improve undoubtedly. But I do not want to go into the ins and outs of how it will improve because I do not like to deal in the abstract. Suffice it to say, though, that with the removal of the capitalist mode of production, and the dissolution of moneyed economies, resources, development, research, clinics and facilities will become much more available. And with the change in economic, social, and political climates and the relations of society, illness will become more socialised, and therein so will sufferers.
Today most of the work that goes into treatment goes into socialising the mentally ill and helping develop the mental skills to manage illness properly. This work is particularly important for men, as by default they are taught in all walks of life to internalise their difficulties. With the awareness of the need to socialise more and more community programmes are being employed, whether its upskilling and volunteer experience for the homeless and unemployed, or sports and swimming programmes for troubled youth, or group therapy, meet ups and peer drive in-patient programmes for the ill. These are the kinds of things that will no doubt expand.
A post-capitalist society would treat all illness as illness, with resources dedicated based on need and providing the best outcomes for the afflicted. It would make every effort to socialise the ill and provide safe and welcoming community environments that placed value on the individual as an expressive, actualised, individual rather than the commodity of capitalist relations.