I believe that universal basic income will result in significantly fewer suicides. I believe this will happen because once basic income is fully implemented then people will have fewer financial reasons to engage in suicide.
However it is important to note that the mind is abstract and cannot be literally sick nor literally well. So discussing mental health as a literal thing is ultimately counterproductive.
Read the essay The Myth of Mental Illness by psychiatrist Thomas Szasz. You can read it for free online. There is also a book version.
Furthermore, it will be good if we can reduce suicides to zero. However, suicide should be respected as a civil and human right for all adults when it is done in private. Read Szasz's book Suicide Prohibition: The Shame of Medicine if you want to learn more.
We should implement universal basic income. However talking in terms of mental health or mental illness is misleading, and is language that is frequently used to justify psychiatric coercion and nonconsensual psychiatry, both of which should be illegal.
Mental health is a metaphor, like calling a joke a sick joke. Literalizing metaphors will have unintended consequences.
Again, we should implement universal basic income; we should simply use concise and accurate language when discussing it. Basic income will likely increase emotional well being of many individuals, and will likely result in fewer suicides. However, the idea of mental health is so subjective and nebulous that using this psychiatric language probably harms the cause of basic income ultimately.