Alzheimer's has such differing effects on individuals, and progresses at such different rates, that there is no specific legal standard either allowing or barring someone with a diagnosis from marrying.
Instead, whether someone with Alzheimer's can marry is controlled by the same legal standard as anyone else: A person must have the same mental capacity as is required to enter a valid contract.
The specifics of this standard are set by state law, and may be slightly different depending on the locale.
In most states, the standard for such mental capacity is whether the person can understand the meaning and effect of the words considered to be the marriage contract. In some states, all that is required is that the person must be able to act in a reasonable manner. And some states measure capacity by the person's ability to judge whether or not to enter into the agreement.
To find out the exact legal standard that applies in the state in which you are interested, do an Internet search of the state and "mental capacity" and "law" and "contract."
As a practical matter, if you or someone else is in this situation and there is some chance that another person will object to the marriage, it may be wise to proceed defensively by getting a formal declaration of capacity. A lawyer with experience in estate planning or elder law should be able to help with this"”although an additional written statement from one or more doctors may also be required as part of the process.