As far as I can tell, people didn't really start believing in germs until quite recently

Even though germ theory dates from the 19th century people were aware that things spoiled and that yeast caused fermentation and so forth.

how did brewers in the old days (or ancient) clean their equipment?

The mashing process obviously sterilizes that wort so thats a big start right there. From there they made sure to cool the wort down quickly to avoid infection. Gruit and then later hop addition to the wort was a big step in adding antimicrobial properties to brewed beer. We dont really see much exporting/commerce in beer until hops become widespread so that points to the idea that hops were effective in increasing the time beer to last before infection. They also lined beer barrels with pitch which is burnt tree resin to create a waterproof barrier between the beer and wood and that too helped protect against infection.

All that said, there clearly could be infection because medieval brewers were very superstitious about he process so clearly the threat of infection was a real life concern.

If you are curious to read more I recommend Beer in the Middle Ages and Renaissance by Richer W. Unger.

This could also apply to other fermented foods where clean working conditions are required--saurkraut, tempeh, soy sauce

Preservation by fermentation is helped by two major techniques: salting and lowering the PH of (making more acidic) the foods in which you wish the preserve. The lactic acid producing bacteria that you find in preserved fermented food such as salami and sauerkraut lowers the PH down enough that other microorganisms have a very hard time surviving.