If so, they make money. If not and the price goes up a bit, some shorts are forced to buy at the higher price to limit their losses. That pushes the price up further, forcing more shorts to buy, etc., causing a "short squeeze" where the price rapidly shoots up very high, "destroying" a lot of shorts.
What you described is options trading, one method of which hedges against declines in stock price. Yeah, one could take a "short" position by buying puts, but it can be argued that a short position is not what puts were originally designed for. In contrast to short sellers who obviously intend for the price to go down, or they won't make any money.
I do question whether puts are the "more common way" given that they have an expiration. You can hold on to a short sell as long as you can pay the interest and the loaning party doesn't want their shares back.