The society is asking for public donations towards legal costs for what will be the first judicial review of a Nice decision.
It is challenging the Nice restriction last year of the prescription of three drugs for people with Alzheimer's. It has joined drug companies Eisai and Pfizer to bring the judicial review.
Neil Hunt, chief executive of the society, said: "Nice is snatching time and quality of life from thousands of vulnerable people who will learn they have this devastating disease this year."
An advert for the campaign, which is running in national newspapers, asks: "Is the quality of life worth the price of a cappuccino? Nice doesn't think so." The advert requests £2.50 from donors to reflect the cost of the drugs per patient per day.
In a statement, Nice said the legal challenge was without foundation.