We (altar boys) would flip the cloth over the altar rail so it would act as kind of a “tablecloth”. There were a couple of parishioners who did put their hands under the cloth, but I don’t think this had anything to do with not having the Precious Body of Christ touch their hands.
In those days we always had an altar server hold the paten (like a plate) under the communicant’s chin to catch the Precious Body in case of accident while placing the host on the tongue.
By the way, several times as an altar boy (in about 5 years of serving the 1962 missal, pre Vatican II) I caught the Precious Body dropped from someone’s tongue. In all masses I’ve attended in the subsequent years, I have seen the Precious Body dropped twice, although I know it happens. That is why I alway receive the Precious Body in my hand, unless a server with a paten is present, then I receive on my tongue.