Well very many of them didn't. A huge number went out of business, to the extent that the total number of breweries in America only recovered to pre-prohibition levels relatively recently. Source: https://eh.net/encyclopedia/a-concise-history-of-americas-brewing-industry/

However, a number of breweries expected that Prohibition would be repealed sooner or later (and they were right, of course) and thought that if they were ready to make beer again as soon as Prohibition ended they'd have a big advantage since so many of their rivals had gone under or at least sold off a lot of stock (and they were right again).

So a lot of breweries just tried to survive and keep their equipment maintained. They couldn't make bundles or money but they could try to wait out Prohibition and be ready to start brewing ASAP when it was repealed.

In the meantime they engaged in a number of side businesses such as non-alcoholic beer, alcohol categorized as medical, soft drinks, etc.

But the most entertaining business they got into was producing malt syrup. This stuff is still used by homebrewers these days (at least those who don't want to brother with actual grains) which you just had to add yeast and water (and possibly hops depending on whether the malt syrup was hopped or not) to at home in order to make beer (although a lot of people added table sugar as well to make it stronger and/or make the same amount of malt syrup produce more beer). Some of them had hilarious warning labels telling people to not add yeast and water and keep it in a cool dark place for a few weeks. It's kind of the equivalent of head shops in which everyone knows what the "water pipes" are for but they manage to be technically legal.

Home brewing was, of course, illegal but it wasn't illegal to sell all the stuff you needed to home brew and it wasn't like police were breaking down random doors looking for a guy making a few gallons for himself. But as a result of this home brewing stayed illegal until the Carter administration and even today an individual isn't allowed to brew more than 100 gallons of homebrew a year (not that that's enforced at all of course).

Here are some pictures of advertising etc. from breweries during prohibition and the sort of things they could still sell: https://prohibition.themobmuseum.org/the-history/prohibition-potpourri/prohibition-products/