Great question!! Quick disclaimer, I'm only speaking for the 19th century American "west" as this is my specialty, so I'm sure others will be able to chime in with facts pertaining to other time periods and regions.

So, a few different tactics were employed, and depending on the culture, region, and individual preference, they were likely used in conjunction with each other to some degree. First off, if you didn't have to go outside, and it was especially hot, you probably didn't. That seems reductive and kind of simple, but the truth of the matter is that, as is the case today, if it was really freakin' hot, and you didn't HAVE to be outside, you probably avoided it. If there was something that necessitated your going outside, or worse, having to work outside, you did it off-hours. By that, I mean you probably got started with your work right around dawn, and then retreated inside or to the shade as mid-day approached. Once dusk began to take hold in earnest, you might go back out and finish up, but only after the sun had really begun to set. You might even wait for a full moon to try and pull a night shift if you had the luxury to do so.

Next up: apparel. If you had to go outside, or work outside, you dressed very deliberately. This meant that you tried to keep as much of your body out of direct sunlight as possible. This meant hats (absolutely crucial), a neckerchief or bandana, thin, long-sleeve shirts, and trousers. It may seem counter-intuitive, but dressing in this manner actually kept (keeps) you cooler than stripping down to your shorts without a shirt, as your body isn't getting scorched by the sun and drying you out. Staying hydrated was a concept that 19th century Americans understood (though not to the extent we understand today), and lots of thought went into assuring a person had adequate water available for any extended periods outdoors. More importantly, you wanted to be sure that your horse had water (if you were working with a horse), and that you knew that established water sources were indeed available, and not dried up.

Humorously enough, many of the things we do today to cool off was also quite popular in the late 19th century (except A/C, of course). People enjoyed ice in their drinks (and it was indeed available in places) as well as ice cream (likewise, surprisingly available in many places that you might not think, like Tombstone, AZ). Baths were also popular.

Really, though, if a person perceived it to be especially hot, they did their best to just minimize their time in the sun: it really is that simple. The people who lived in places like modern-day Arizona, Utah, and New Mexico understood the dangers of dehydration and exposure, and respected their environments enough to minimize their risks in most cases. When it got REALLY hot, like 100+ degrees, people generally just stayed inside if they could, and tried to endure it. When that wasn't possible, they dressed appropriately, worked outside during the cooler parts of the day (or at night) and were careful to stay near (or bring their own) water sources.

[Sources - Bob Boze Bell, 'The Illustrated Life and Times of Wyatt Earp'; Michael Denning, 'Mechanic Accents'; Richard Slotkin, 'Gunfighter Nation']