There was a thread very recently on a similar topic, but I can elaborate a little.

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While there may have been people in the past that cared about preserving antiquities (Ashurbanipal's museum comes to mind), the modern concept of archaeological preservation is not the same.

The idea that looters destroy and damage sites is based on the modern archaeological perspective that artifacts (and the sites themselves) should be treated as scientific data to understand the past. Modern archaeology recognizes that the majority of the information encoded by artifacts is not based on the artifact itself - a lone bowl or spear for instance - but rather in the context of the artifacts: what other artifacts they are found with, where in the site, etc.

Because of this, modern archaeology recognizes that excavation is inherently destructive because once you remove something from the ground you can never return it to its original context again. If the context is where all the information is, then you need to be very careful to record the context as best as you can while excavating. Looters typically do not do this, so we say they "destroy" sites because all that contextual information is lost.

So in sum, our modern distinction between looting and archaeology is based on the perspective that archaeology is a kind of science and that artifacts and sites can be used to systematically study the past. On the other hand, the antiquarian perspective (often shared by looters) has preservation as the only goal, as opposed to the generation of data. Then you have treasure hunting, which views artifacts as a method for generating wealth.

All that said, to answer your question, like so much of modern science these ideas came out of enlightenment thinking. In the United States, Thomas Jefferson is widely credited as conducting the first systematic excavations. As in, he actually took notes on the context of the artifacts and the sequence of construction of the mound.

More generally, developments in geology at the same time were a huge influence on the field of archaeology (although not for some time later). Geologists in the late 17ths through 19th century like Nicolas Steno and William Smith helped develop the concept of stratigraphy which places importance on the relative position of different geologic layers with each other. Basically, this is the origin of the archaeological focus on context instead of just on individual objects.

As far as perceptions in the past, they can vary tremendously from the antiquarian perspective ("It belongs in a museum") to pure treasure hunting. Ancient Egyptians looting tombs of Pharaohs and other elites is one example, and the mining companies set up by the Spanish in the Andes to extract gold (in the form of gold objects) from archaeological sites (mainly burials) is another. The former was likely not a profession, but rather somebody taking advantage of a situation that presented itself. The latter, on the other hand, was genuinely a profession in the sense of groups of people applying for mining permits from the colonial government to mine for "gold" the same you would stake a claim to a vein of ore or an oil well. In general though, I imagine most looting in the past was opportunistic and focused on valuable raw materials (like gold) rather than modern looting which focuses on artifacts which have value as art objects. That market didn't really exist in the past to any huge extent, so making a living out of it would be difficult.