Is it time to put her to sleep?

My cat is 16 years old She does not groom herself well and lately after using the liter box she scoots across the floor rubbing her bottom and leaving a mess on the floor. I have to clean her often cause she has caked shit on her. Is this a particular health issue or is it just her age? Is this an unhealthy situation for children and adults in the home? She is other wise healthy,eats well and drinks lots of water. Sleeps a lot and does vomit often.Is it time to put her to sleep? I have been keeping her confined to a room to keep her from making messes elsewhere in the house. She is a sweet lovable cat that loves to be on laps.

Since she's eating & drinking normally, no, it's not time to put her to sleep. Cats who are dying don't eat & drink properly & hide all the time, looking for safe places.
   
It is very likely a specific health problem - maybe consistently soft stools, from diabetes or any number of other problems; even a weakness in her muscles. Take her to the vet for a review of her digestive system, especially, & a full physical & panel of blood tests.
   
The grooming problem could be a side-effect of her health problem.
   
Yes, there could be a health issue for the humans, as some cats do harbor parasites that can affect humans, so it's best to confine her until she's been treated.