People were very much aware of the disease transmission since antiquity, what included the transmission through direct or even indirect human contact. As you can read in the responses by u/J-Force and me in a recent thread concerning similar topic, the main development of 'germ theory' was the discovery of actual pathogens that was possible thanks to the development of optics and was not entirely complete before the in-depth research of the viruses in the second half of 20th century. In other words, the main discovery of what is commonly termed 'germ theory' was the fact that the many diseases are caused by a microorganisms that can be targeted and destroyed thus facilitating healing. This does not mean, however, that such an explanation of the transmission of diseases was not suspected or intuitively accepted by many long time before the formulation of the modern germ theory.
The discovery of microorganisms, although groundbreaking, was for the large part only confirming what people were suspecting or even taking for a fact while not being able to precisely explain. The idea that at least some diseases might be transmitted by some non-observable contagious vector can be found in Galen's writings and evidenced by various measures people in the past were taking to minimize possibility of the disease spreading. The earliest examples of such measures are found in the Bible (mainly in Leviticus), where the rules of conduct include mandatory destruction of all vessels from a sick person was using. Similar measures were taken in Middle Ages, as the possessions of lepers and victims of bubonic plague were customarily burnt and people handling corpses of the diseased were usually trying protecting themselves to minimize contact and thus decrease chances of contagion. Sequestering of the victims of leprosy (it is worth noting that this term could have been applied also to visually similar skin conditions) and bubonic plague was also a common way to decrease possibility of contagion. Dukes of Milan, Gian Galeazzo Visconti and Galeazzo Maria Sforza officially mandated quarantine and sequestering the sick in the 1380s and and 1460s respectively. Similarly, in 1497 James IV issued the Grandgore Act mandating all people suffering from pox to be sequestered on the island of Inch Keith under the penalty of banishment. A year earlier, city council of Nuremberg issued a ruling that all people suffering from 'pox and plague' (presumably any rapidly progressing illness with easily visible symptoms) to be quarantined in the Holy Cross Hospital outside the city walls and that non-residents showing signs of such diseases were to be denied entry into the city. There are also known instances of member of organizations being asked to not come to the meetings until making sure that they are healthy, as in the case of Frankfurt councilman Klaus Schellen whose wife and kids contracted pox some time in 1502. Furthermore, aforementioned councilmen of Nuremberg wrote an inquiry to their colleagues in Bamberg asking whether the recent outbreak of pox could have been caused by the infected meat.
This was also nothing new. Marchione di Coppo Stefani notes in his Florentine Chronicle (rubric 643) that people were afraid of entering houses where people died, that death was common among gravediggers and people assisting in burials and that the city was essentially in lockdown, with guild ceasing to operate and all taverns and shops (save for apothecaries) being closed. It is interesting that he also mentions that people were leaving the city, but he adds that 'where there had been no [pestilence], there they carried it'. It is very likely that plague was as dangerous as it was because people were taking measures that were efficient against anthroponotic contagion, unaware that bubonic plague had a significant zoonotic vector, as it was largely (but not solely, according to modern research) transmitted by rats and fleas.
The concept of contagion caused by an exogenous factor was quite common in Middle Ages and in early 16th century it has been developed by Girolamo Fracastoro into the theory of the 'seeds of illness' that could have been transmitted through direct contact, indirect contact (e.g. usage of the same items) and 'through air' what has been detailed in the book De contagion, contagiosis morbis et eorum curatione libri published in 1546. In late 1490s, Alexandri Benedicti from Venice wrote on the first examples of syphilis noticing that the disease might be transmitted through sexual contact. His contemporaries, Spanish doctors Gaspare Torella and Pere Pintor were also discussing pox and syphilis at length, pointing to the possibility of the transmission via air or direct physical contact, including sexual one.
As one can see, all the examples presented above clearly show that people as early as 14th century (and quite possibly also earlier) were pretty much aware on the various vectors of contagion even though they were unable to precisely explain them.
Arrizabalaga, J., Henderson, J., French, R., The Great Pox: The French Disease in Renaissance Europe. Yale University Press, 1997.
Carmichael, A.G., Contagion Theory and Contagion Practice in Fifteenth-Century Milan, in: Renaissance Quaterly 44, no. 2 (1991), pp. 213-256.
Lopez-Piñero, J.M., The medical profession in 16th century Spain. in: Russel A.W. (ed.), The own and state physician in Europe from the Middle Ages to the Enlightenment. Harrasowitz Verlag, Wiesbaden 1981.
Nuremberg State Archive, Verlässe des Innern Rats [Acts of Inner Councils], no. 336, fol. 7v-9r.
Rodolico, N. (ed.), Cronaca fiorentina. in: Rerum Italicarum Scriptores, vol. 30. Citta di Castello: 1903-13.
In Euro-America in the Early-Modern period some disease theories had to do with miasma or bad air. That nasty smells and substances in the air would cause illness. This explains why people put flowers in the plague masks. There is also an intuition of the infectiousness of disease that (likely) most people had. Everyone has seen a cold go through a house. Germ theory is crucial to a detailed understanding of how disease spreads but it is easy enough to intuit that disease does spread between people and through objects and animals.
That said, and this reaches slightly beyond the scope of your question, it’s always important to remember that.People weren’t stupid. People knew, qualitatively, how the body worked (this was the pre industrial place of witches, wise-women, and midwives) and had a rough knowledge of what to do in the case of a fever and also how to stop a disease spreading.
Disclaimer: I am an historian but I study German intellectuals in the 19th century so this is out of my sub-field and it should be taken with a grain of salt. Further topics to explore for early epidemiology -miasma -John Snow v. Cholera -Folk Medicine -Anything in ancient/medieval plague studies
In Lindsey Fitzharris's book "The Butchering Art", and David Orshinsky's "Bellevue: Three Centuries of Medicine and Mayhem at America's Most Storied Hospital" the belief in the pre-germ theory age of medicine was that disease spread from Miasma. Miasma was taught and believed to be a foul air or mist that came from decaying matter and was usually denoted by a foul smell.