I would contact the owner of the unit via letter and advise them of the situation clearly indicating the boards concerns and for example; tenants violating house rules, tenants doing this or that. I presume your issues are with the tenants (that are managed by the management company). And I also assume that the owner hired the management company.

I agree with Mark and Ned, however there are some additional things that can be done if the Board is in agreement.
1). A Board letter to the Holder of Unsold Shares (Sponsor) letting them know their tenant is in breach of the House Rules, or Bylaws, or Proprietary Lease - whatever are involved;
2). The same letter from your Management Company to the H-of-US;
3). followed by a letter (when nothing improves) from the Board's/Coops Lawyer with a demand to correct the situation.
4). Then the Board authorizes the attorney to issue what is known as a 'Notice To Cure' document. This establishes a set of demands, based on legal criteria, and a reasonable deadline (relative to the nature of the issue) for the 'Cure' to occur by, if it doesn't, the Board can then move to force the divestiture of the apartment legally from the shareholder.
Because, the Sponsor is still a shareholder, and while they have some special exclusions as to sales etc. they are still a shareholder and you have a relationship with them, and not the tenant. That is your point of attack. It can work. Don't let his Management Company who handles property for him screen or buffer the shareholder from you, they don't actually have those rights due to the fact that they are not the shareholder, the Sponsor is. However, be prepared for verbal promises, assurances, all stall tactics. start a paper trail, and insist that all communication be in writing. It's not fast, but whether they admit it or not, they can actually lose ownership of the shares. None of the above applies to Condo's, only to Coop's.
By the way, if it is a Proprietary Lease breach, and they have really angered a very large portion of the Shareholders, you can do super-majority vote of Shareholders to force the sale. and that (Prop. Lease Votes, not Bylaws- that has to be at a specially called meeting) can be done by a mail vote based on a carefully and legally written motion.

Writing to the tenant is problematic because he has diminished mental capacity, possibly early dementia, and has an executor handling his affairs.
The issue is the tenant being unable to care for himself and being in an unsafe situation.

You don't actually have a legal relationship with the tenant, you, as a Coop/Board have a relationship with the shareholder of the shares related to that apartment. You therefore have to direct your efforts to dealing with the shareholder, who is answerable for and responsible for, ultimately, the behavior of their tenant(s). They are in a bad position given the rights of Rent Stabilized Tenants under the law, and the Court being very, and appropriately, protective of the disabled and seniors with possible dementia issues. Is there any family with responsibility here? A guardian? You might need to contact social welfare services etc. in the process. IF there is anything that is occurring that is a potential endangerment of that person to themselves or the other shareholders then you need to call the Police, or Fire Department, especially to get it on record. Do not enter the apartment at all, if there is an emergency and the Board has rights under those conditions, there must be at least one Board member to accompany the Super, and it is better if a Police Rep or Fire Dept Rep is with you, as they have legal protection that you do not. IF you enter without them and there is a problem, or question as to the 'realness' of the emergency, you can be in a lot of trouble.
The most important thing is to start a paper trail.EVERYTHING in writing, return receipt, and consistent, do not let time lapse as the Court will view it as not current or just occasional. Your best shot is to leverage the holder of unsold shares who has no protection from you writing and filing for breach of Prop Lease, Bylaws, House Rules, whatever is affected. You must keep an accurate log of times, dates, incidents and who reported to. Be proactive, issues do not usually go away by themselves, they tend to progress and deteriorate over time. Be kind, but be firm. You have an entire Coop to protect. It is best to involve your Coop attorney early in the process if you feel it is not going to be a simple matter.

Unfortunately, there isn't much that you can do from your side to affect the management company that the original sponsor hires for their individual units. However, if you do feel that things are neglected or are not properly being handled, I would document them and send them to the sponsor that has hired them. Depending on the level of involvement, the sponsor may or may not be aware of the deficiencies that exist with their current agent and may look to a solution if there is negligence found (or if these actions are negatively affecting their value).

More likely than not, the sponsor is losing money on the apartments now (the income being received on these apartments may be less than the maintenance that they're paying each month) so they may not be too keen on hiring another firm that may cost them more money out of pocket.

Writing to the tenant is problematic because he has diminished mental capacity, possibly early dementia, and has an executor handling his affairs.
The issue is the tenant being unable to care for himself and being in an unsafe situation.

All the more reason for the board to write to the owner; it is a letter of concern for the tenant and their capacity to take care of, both from and a health and safety perspective, themselves (and the unit) and that the board has a responsibility for the health and safety of residents. I would cc: their management company.

The owner needs to be advised of the boards concern for their tenants welfare. It's in the owners ballpark to determine the best course of action (initially). I am certain that a letter of reasonable and compassionate concern will prompt the owner into action. Give the owner a chance to address the concerns before you take radical action as suggested by dsil. Any court would ask "did you make the owner aware of your concerns?".