How was anxiety viewed prior to the invention of modern psychology? How was it treated?

I can answer this question in a North American context, but the mileage of my answer may vary depending on the time and place you're wondering about.

The use of the term anxiety (disorder) to identify a mental illness is fairly recent, but "anxiety" was often listed as a cause of insanity. I specialize in the nineteenth century, so I'll limit my answer to that period and let experts on other periods fill in those blanks!

I'll use the Toronto asylum as a reference point (it's my personal specialty and thus the place with which I am most familiar). Anxiety comes up frequently in the register as an "exciting cause" of illness, but not a diagnosis per se. Here are a few excerpts from the files of people for whom "anxiety" was listed as an exciting cause:

William Mosly, 1847, "fear of fulfilling his promises"

Elizabeth Clark, 1845, "anxiety about son" and "fear of fire"

Phoebe Thistle, 1844, "anxiety about daughter and husband's illness"

I could go on, but that provides a decent snapshot about how anxiety was identified. As for how it was viewed, it was generally a matter of some concern when worry became so great that it began to interfere with a person's daily life. At such a point, that person might be institutionalized. The reasons why vary depending upon your school of historical thought. Those who subscribe to "social control" theory (which can usually be traced back to the writings of Michel Foucault and David Rothman) will suggest that extreme anxiety and worry was identified as a problem when it became a significant disruption to daily life, and those with extreme symptoms would be removed to an asylum to prevent further disruption. Thus the asylum was an institution of control, and anxiety was a threat to order. This is, of course, the extreme "social control" stance, as it has been tempered over the years. The other extreme camp, sometimes termed "meliorists," would argue that anxiety was viewed (at least by asylum doctors) as a very curable illness under the popular "moral" theory of mental illness. If a patient was removed from their unhealthy environment, they could convalesce. Thus, anxiety would have been viewed as an unfortunate but curable affliction.

Realistically, the truth is probably between these two extremes. Families may have found someone with extreme anxiety inconvenient and thus sent them to the mad-house. But they may have also found their relations' aberrant behaviour concerning and committed them to the asylum under assurances that they could be rehabilitated.

It is also worth noting that theories of mental illness such as moral theory were in their infancy in the mid-nineteenth century, so scientific knowledge of mental illness would not have been nearly as widespread as it is today. But, theories of the mind and insanity were well-enough understood that someone's first thought upon encountering symptoms of anxiety would probably have been something to the effect of "Oh no, my mother/sister/daughter/wife is going insane." It could have been viewed as something of an embarrassment, while simultaneously a cause for concern. Those with mental illnesses were still loved by their families, who often withdrew them and readmitted them from asylums many times in their desperation to have them home and well again.

A little more on anxiety:

The term "anxiety" would not always have been used, but similar symptoms would crop up in asylum registers under several other names, depending on the context. For instance, the Toronto asylum saw many admissions for "religious excitement" in the 1840s, probably due to a bunch of millenarian predictions of the end of days in that time. The symptoms listed in the asylum register resembled what we might today call "anxiety": excessive worry, catastrophic thinking, what we would call "anxiety attacks." Another common reason for admission was "melancholy." It's easy to assume these patients were suffering from what we would call depression, but many of the symptoms listed also resembled modern anxiety disorder.

EDIT: I forgot to really address treatment. Moral treatment involved a combination of removal from the affecting environment and a "rationalizing" routine of work and leisure. Realistically, some asylum doctors would have also employed methods such as the shaving of the head, "cupping," and in some cases (though it went against the principles of moral therapy) restraint.