Incontinence alone is not a good reason to put a dog down or put a dog to sleep.
But if the dog is having other health issues and cannot walk very good and is not having a good quality of life and has severe incontinence then putting the dog to sleep would probably be best.
If the dog is just incontinent and has bladder issues then you can help the dog by using disposable dog diapers that have a hole in the diaper for the dogs tale.
The dog diapers are just like baby diapers and adult diapers and tape on them and then you change them when they are wet.
So if the dog will wear dog diapers then you could try that.
If the dog doesn't want to wear the dog diapers then you either have to deal with the urine on things such as dog urine pads or put the dog to sleep.