For 1 day in cozumel from the cruise. I heard car rentals are expensive after getting the insurance. But if you take a taxi, do they have insurance on their passagers?

We NEVER rent a car in a port! You don't know the roads, can't look at the scenery and the drivers - especially in Cozumel - are wild & crazy. You can get a cab to take you anywhere and/or have them give you a tour of the island as we did on one stop there.
If you are going to rent a car call your insurance company and find out if they cover you in Cozumel.

If you rent a car in Mexico, you really need to buy insurance there.

I'm not sure what you are talking about with regard to taxis having insurance on their passengers? Do you mean liability if they have an accident? Presumably there is some kind of liability coverage if the taxi is licensed, but I've never had that experience so I can't comment directly. I'm not quite sure why this is an issue for you. In all my years of traveling it's not something I've ever really worried about since I do have health insurance if I have an accident.

I would never drive in Cozumel for a day visit. Take a registered taxi from the stand. They have set prices for the places they go.

Are you going to another country without health insurance?