Questions like this are deceptively tough to answer, because it rests very much on the subjective judgment of what "being hot" means, and how people think to deal with it. We talk about heat like it's something objective, but it very much isn't! What I might find oppressive in upstate New York, someone in Florida might consider a nice day. What made John Adams very uncomfortable might have hardly bothered Thomas Jefferson. So therefore how to handle heat is going to change depending on the person.

But there are some across-the-board statements we can make. The basic thing people did was shift to clothing more appropriate for warm weather, which by the nineteenth century generally, though not exclusively, meant wearing cotton or cotton blends in paler colors and lighter weights. For men, this might mean white duck trousers (a very common part of summer military uniforms at the time) with unlined coats; women might wear dresses made of a lightweight printed cotton, or some sort of sheer or semi-sheer cotton made only partially lined. By the late nineteenth century, manufacturers were providing corsets designed for the summer, with a stiff net used for the body fabric or eyelets punched into it to allow for air circulation. Summer bonnets and hats would also be worn, usually made out of breathable materials like straw and with a wider brim, and even a lace or net veil could be use to protect the face. And of course, parasols would also be used to provide mobile shade.

I wouldn't use the word "taboo" to describe nineteenth-century Western views on showing legs and arms, as that implies a stronger distance from modern norms than I think really is appropriate. Something I keep going back to in discussions of historical fashion is that it's standards for what is formal vs. informal that have changed more than anything else. For instance, in the period, formality was the norm and informality would be something you only indulged in at home, whereas today we expect to be relatively informal unless working in an office or at a special event. Shorts for men are considered inappropriate for formal dress today, but there are more situations today where men can be informal in the summertime.

(The calculus for women was/is different, since women frequently are and were expected to show more skin in formal dress. I would note that in the first decade of the nineteenth century, short sleeves were fashionable and could have been worn in any situation, and that in the second half of the century we see fashionable semi-formal forms of dress incorporating elbow-length sleeves. Servants and poor women were welcome to roll up their sleeves whenever they wanted or needed to, as well.)

It's important not to assert that things modern reenactors do that help to keep them cool were therefore necessarily done by people in the periods they portray, and that how they feel about their clothes in the summer necessarily reflect how the latter felt about theirs, but it should still be noted that many reenactors do find that breathability of the fabric is more important than actually baring their skin. It's been said many times (and I've experienced it personally as well) that having the skin shaded by fabric is more cooling than having it bare and exposed to the sun, and that staying out of air conditioning makes entering the heat seem less abrupt - which goes back to my initial point about the subjectivity of the concept of heat.