There's no conclusive evidence that folic acid supplements improve cognitive function in older adults or in people with Alzheimer's disease or other types of dementia.
Blood levels of folate are classified as either low or normal in the general population. Low folate blood levels are associated with poor cognitive performance, which could be improved by folic acid supplements. However, folic acid is not helpful for people with normal blood levels.
One study showed cognitive function was significantly better in one group of older adults who took folic acid supplements than in those who did not. However, the participants were already at some risk of cognitive decline because of deficient blood levels of folate. So the results of this study can be generalized neither to all older adults nor to those with Alzheimer's disease.
A review of eight randomized, controlled trials found that the use of folic acid supplements has no benefit on cognitive function in healthy adults or in those with mild to moderate cognitive decline or dementia.
So although it doesn't appear that a folic acid supplement would benefit everyone, it may be something worth discussing with your doctor. Keep in mind that in the United States many foods such as breads and cereals are fortified with folic acid. If you're at high risk of developing dementia or have already experienced some cognitive decline, checking your folic acid levels may be a reasonable step.