Or when did men stop carrying them?

Carrying bags has always been common. They weren't, however, always functional in the same way as purses. They might carry sewing or perfumes, for example. Under their skirts women had hanging pockets to carry the more valuable items. Pockets go in and out of style depending on how fitted the skirt of the gown is. While very common in the 18th century, when the slim, sheer gowns become fashionable around 1800 you can't put a pocket underneath that won't show. Bags become more common. They fall back out again as the skirt size grows and return at the end of the 19th century. Womens garments today, as throughout the 20th century, are too slim and fitted to be able to fit things into pockets without bulges, so purses are the answer.

Men have carried bags for many reasons, just like they still do, but they've never been called purses. Haversacks/backpacks, knapsacks/messenger bags, market wallets, drawstring sacks, etc.