Who would be such an idiot to claim they're "safe" in the first place? Any investing is never "safe": you can always risk losing part of your money, even if you invest in a house. Not investing your money isn't safe either.
Indexes can certainly dip for years: that's why you manage your risk and only invest money you don't need that much so that you don't have to cash in your investment during bad times. As long as the economy grows the growth will eventually be reflected in the indexes. Even a single 10-year period with a common stock index lower in the end rather than in the beginning would be, in practice, a rare occasion. Most depressions are over in a few years. Some continents can tank longer but you will want to buy indexes from different areas of the world anyway. Or you can only buy your local indexes, making sure you will yourself go up and down with the indexes you buy.
An index will grow a few percent annually in the average, compounded. This is over decades which imply bad years and very good years. To buy the most passive index funds that have the smallest operating costs and letting your money sit in there is more and more likely to beat active funds and individual stock picking simply by waiting for the years to go by. Any costs saved by skipping active management fees or stock transaction fees will accumulate into your earnings instead.