In modern times people keep beer from spoiling by utilizing proper sanitation and hops (which are a great preservative). However in ancient times people didn't have access to hops which makes things significantly harder. There are, however, some other ways ancient brewers could respond to spoilage. Which were used would vary widely from time to place.

1. Just let the beer spoil: spoilage in beer is generally caused by bacteria. Some kinds of bacteria are intentionally used in sour beers in modern times because they taste good so letting the beer spoil isn't the end of the world. This is especially the case before the advent of hops. Hoppy bitterness is balanced by malty sweetness but beer without any hops would've been often been cloyingly sweet so a bit of bacterial tang could've been appreciated by some people (depending on the kind of bacteria, lacto bacteria are going to be much more pleasant than aceto bacteria for example). Some recipes for archaic forms of ale such as West Country white ale that were recorded would have pretty much inevitably lead to the presence of bacteria and a lot of (but by no means all) farmhouse ale that's still brewed in out of the way places in Europe has at least some bacteria present.
2. Drink the beer before it spoils: generally speaking, yeast eats sugars faster than bacteria but bacteria is a less picky eater. So if you put both bacteria and yeast in some wort (unfermented beer) and then sample it repeatedly as it ferments it'll start off cloyingly sweet and then the sweetness with quickly drop as the yeast gets to work but for quite some time you won't be able to taste much bacterial sourness, but if you wait long enough the sourness will come to dominate the beer. So if you time things right and drink the infected beer after the yeast is mostly done but before the bacteria has really taken over, you'll be able to get beer that doesn't really taste sour or spoiled. This was very much easier in pre-modern times since you'd just ferment the beer and the drink it fresh. In modern times brewers have to ferment the beer, then bottle it, then ship it, then have it sit on shelves, then have people bring it home and then drink it. This means in modern times you have to have beer that won't spoil for a good long time. But in pre-modern times if you're going to drink it FAST (especially if you brew it fresh for a specific event such as a wedding feast) then it doesn't really matter if it's going to end up spoiling relatively quickly, you just drink it before that point. A good example of how this works is Korean makgeolli (Korean rice beer). To make makgeolli you put in both bacteria and yeast and then historically you'd drink it once the sweetness, booze, and bacterial sourness all balance each other nicely. However, this historical kind of makgeolli can't be sold commercially since once you hit the point where you get a nice flavor balance the makgeolli doesn't STAY there and continues to ferment and becomes really dry, sour, and unpleasant after a while. To make it more commercially viable, most modern makgeolli is brewed with less bacteria and more yeast, allowed to ferment dry, and then back-sweetened with artificial sweetener. This isn't as good, as you can imagine, and you can still find the old stuff that just doesn't have any shelf life but it's very much the minority. However, in pre-modern times a lot of other places aside from Korea certainly would've had a "drink it up fast before it spoils" approach to beer.
3. Use various herbs: hops are an amazing preservative because they inhibit bacteria but don't bother the yeast much. There are plenty of other herbs that can keep out bacteria but most of them make it hard for yeast to do their thing, or they let the yeast do their thing but don't do a good job of keeping out the bacteria. There were herbs and herb mixes that were used for this purpose, the most famous of them being gruit in the Low Counries, but they were often more expensive and didn't work as well as hops.
4. Uses the right yeast strain: certain kinds of yeast are more helpful if keeping out bacteria is a priority. Before pure yeast strains were isolated by the Carlsberg brewery in the late 1800's brewers yeast was often a mix of many different strains and even species of yeast. For example Norwegian yeast strains (generally called "kviek" because that's the Norwegian word for yeast) often ferment beer very quickly. This is helpful as it gives the bacteria less time to get a foothold before the yeast eats most of the food and creates alcohol (and alcohol itself is a good preservative against most kinds of bacteria). Belgian saisson yeast strains, meanwhile, ferment beer so incredibly dry that they don't leave much food left for bacteria. Brett species of yeast (used historically mostly in Britain, hence the name, before surviving mostly in Belgium after mostly dying out in the UK), while often slower fermenting than normal yeast, also do a great job of drying out the beer and leaving behind less food for the bacteria. Brett yeast also does a great job of eliminating dissolved oxygen which is a good thing for preserving beer. However, saissons and brett yeast have very strong flavors that some people (including me) find unpleasant, while kviek (in my experience) tastes like sunshine and love. Lager yeast also ferments well at lower temperatures and low temperatures are great for inhibiting spoilage but lager yeasts probably didn't exist in ancient times and they seem to be (bizarrely) a hybrid of European brewing yeast and South American wild yeast and how the hell it ended up in German beer nobody knows.
5. Make it strong: alcohol is a pretty good preservative. Stronger drinks like wine don't need hops as much as beer does which is part of the reason that people put hops in beer but not wine. So if you make a good strong beer that should help prevent spoilage. The only issue here is that the sort of carbs present in wort (unfermented beer) are generally more complex and harder for yeast to eat than the carbs present in must (unfermented wine) so there'll often be more leftovers that can be eaten by bacteria in the fermenting beer than the fermenting wine. Also, beer yeast tends to have lower alcohol tolerance than wine yeast. So there are more hoops to jump through but in the past if you wanted to make a beer that wouldn't spoiled over and extended time you'd generally make it strong.
6. Be careful: historically brewers had a lot more practice than modern newbie homebrewers. If you make beer over and over and over and over and over with a lot of coaching from an older relative who's made beer countless times you're not going to make the soft of newbie mistakes that result in infected beer. What some people did was keep some yeast down the well where it was cool and if you ended up with an infected batch, you'd borrow some yeast from a neighbor instead of using yeast from the infected batch. Once you get the hang of it, it really isn't that hard even without modern sanitation techniques and a lot of experienced homebrewers in modern time get pretty lazy about sanitation and still go for years without getting an infected batch. The process of beermaking itself (mashing) requires high enough temperatures to do a good job of killing bacteria even if pre-modern brewers very often didn't boil the wort (boiling isn't as important if you're not using hops). Personally the last time I had an infection was years ago when my fermenter started leaking so I put a bucket under the fermenter to catch the drips and the open bucket (obviously) got infected. The infected beer was actually pretty good, nice tangy sourness with no off flavors.