This can be a complex issue and the answer will likely change over time.
If the individual is still able to adequately care for the pet and is not at any increased risk because the pet resides with them, it may be a very good idea to have a pet. The benefits of having the love and companionship of a pet contribute to our quality of life. Some pets can provide some degree of added safety if properly trained, and may help provide reassurance to the individual and to their family members. I have been reading studies about using interaction with pets to help elicit greater compliance to requests for bathing, eating, taking medications and more in those with dementia-related illnesses. Some of what I have read suggests that because our pets make us feel good, this has a biochemical effect on us. When we are happier, calmer, feeling loved and loving our pet, we are more positive and compliant. The same appears to be true for many people with dementia and Alzheimer's and their interactions with pets -- both their own and "visiting" pets. Some studies relate a direct correlation between increased mood, calmness, and compliance with interactions with a pet. Another study related an example of an increase in verbalization due to interaction with both pets and talking stuffed animals or "pseudo pets".
Challenges do occur when the individual is not able to properly care for the pet. The pet's health might suffer as a result. A common challenge occurs when the person does not remember or is physically unable to let the pet out several times a day to relieve itself and instead the pet relieves itself inside the person's home or apartment creating an unpleasant and unsanitary living environment for both the person and pet. An unfortunate and unintended risk could be introduced if the pet were to cause the person to fall if the pet is not well-trained.
Finally, the cost that comes along with owning a pet may put a strain on the individual's monthly income and should be taken into account as well.
Whatever the initial decision is regarding a pet, the situation should continue to be monitored regularly as things will likely change over time.