The "old husband's tale" was once valid, but is not any longer. Not that many years back, the standard container for lead-acid batteries was hard rubber. It was a good insulator and resistant to acid, and had the benefit of some resilience so the cases didn't crack or split at the seams. You might, with some luck, find an "antique" hiding in someone's garage and be able to inspect the casing of one of these beauties yourself, though most of yesterday's batteries have long since been recycled at the local lead smelter.
The danger of concrete was not electrical in nature but rather chemical. Lime (calcium) in concrete or cement could pass through the hard rubber compound whereas the acid could not. Cold wet concrete, not unusual for a garage or driveway, set up the conditions for the lime to actually leach into the battery casing, and it would, aided (I think) by the charged ions of the electrolyte egging it on. Battery activity requires a certain concentration of the acid, measured by its specific gravity. The lime would neutralize some of the acid, altering its concentration and producing a salt which would pollute the plates of the battery. The electrolyte would have to be purged and the salts washed out before you could put fresh acid of the proper concentration in-- just a pain. Further, because the lime got permanently into the rubber itself, it was less resilient on the bottom of the battery and was more susceptible to the same pollution at a later date. Hence the battery manufacturers would warn the service-station and garage owners to always rack their batteries and not store them on the floor once they had been "filled".
The military were the first to use acrylic and pvc cases for their batteries, which were often filled and ready for use many months prior to their actual use. When good strong pvc cases were economically feasible in the mid '50's (think Tupperware), the battery industry moved quickly away from rubber and into plastics. It was a good move, being at once cheaper and solving the lime-leach problem. The cases are also lighter and don't tend to expand or contract as much, important because the seal between the case and the terminal posts, which used to be a real bugaboo, is now truly tight. And the plastics transfer heat to the outside of the battery better (though now, with today's tight engine compartments, a battery over-jacket of foam rubber is sometimes needed to protect the battery from the heat of the engine!).
How long would it take for a rubber-cased battery's electrolyte balance to be upset (to the point where the battery's life was significantly affected) by lime leaching in? I have no direct experience, but would guess that a couple of months would be enough. But long before that, you would probably be able to measure a voltage drop in the cells, due to a less effective concentration of electrolyte-- possibly after a week or two. In the battery biz, where you have to take ambient room temp into account when measuring a cell's specific gravity, where .5v differential between (nom.2.2v) cells can be a cause for concern about the battery's health, little things can cause a big spoil.
Some of you old-timers in the telcom industry will probably remember glass jar casings for some of your huge central office batteries. They were taking no chances back then...