How do I proceed in opening a day care for seniors?

Before you get any farther or do anything, check with your local community to determine if the zoning regulations for the area in which you want to start a day care center allow commercial operations.    

If not, you'd have to apply for a variation, which you may or may not get. 

Wouldn't hurt to check with the community's tech division as well; they typically have a lot of regulations.

This will be a BIG undertaking. Unlike a child daycare, you may have to have CNAs and a Registered Nurse affiliated. A lawyer will have to help u with paperwork.

Your state of Utah regulates adult day cares. Just like with a child day care, you have to be licensed. You should read this https://rules.utah.gov/publicat/code/r501/r501-13.htm

I would start with the State and see if there are regulations for Senior Day Care.
If you find nothing contact the State Ombudsman they should be able to point you in the right direction.
Another call to make might be your local Health Department or the County Health Department.
There are codes for Day Care Centers but those are usually Child Care but not sure if they would follow the same code for Adult. I would imagine the difference would be the Staff/participant ratio.

another good place to start would be contact a local Senior Services and see if they are aware of any regulations.

And There would be a big difference if you are providing showers if they are needed. Will you be changing soiled briefs or would you call and have a responsible person come pick up the person like they do in some preschool programs. (If you do not plan on changing soiled brief/pull-up's then you should limit the time there to no more than 3 or 4 hours)
Will you be providing lunches? breakfast? if so catered or making them in house? If you make them you will need an approved kitchen (Health Department regulations will have to be followed) Someone will probably have to be a Certified Food Handler (again Health Department regulations)
And maybe the first person you should talk to is an insurance Agent to determine how much the insurance will cost. I would imagine it would be very expensive.

I would go to some you know and discuss them with them. They are the ones in the know. I will tell you that regulations and liabilities will be ONEROUS at best and that families can be a trial.Speak with people first; in fact consider either volunteering or working in a place first. And good luck. This is so needed and is a vocation. Keep us up on your progress.

My cousin ran a nursing home for a little while in Texas. He could not get over the amount of regulation they were subject to and how difficult and burdensome the oversight was. Even though you sound like you view this as a non-profit or mission, the state will view it otherwise and will oversee you like a highly-regulated business (much like a children's daycare). It sounds like an awesome pursuit. My DH and I have run a business together for 30+ years and I won't romanticize it to you, especially when you have to deal with employees and contact with the public, so please go into this with your eyes wide open. Wishing you success!

Check with state and county licensing agencies.