Not just the Middle Ages but this belief existed from Greco-Roman times all the way into the early-modern period and the advent of germ theory (and even beyond). I've quite extensively studied plague in the 16th and 17th century, so most of my examples relate to that, though I feel they're indicative of wider beliefs from the period.
I can't comment on disease theory outside of Europe, but yes, bad air or 'miasma' was indeed the accepted, and generally unquestioned, consensus on what caused illness.
Pre-1500 or so disease theory was much more steeped in superstition and religion, particularly the notion that pandemics were direct and instantaneous visitations of God's wrath. Moving to more towards the 17th century though, disease theory had become more grounded and practical, and people were beginning to look at disease vectors and how the body became infected.
Outside of quarantine procedures, nearly all London plague orders relate to miasma and purifying the air. 17th century public health in England was in a fairly poor state, worsened by rapid urbanisation from 1500 onwards. Plague orders routinely singled out butchers as being a major polluter, particularly open-air slaughterhouses and the practice of dumping offal and blood in the streets. Though poorly enforced, 1607 ordinances to the Butcher's Company forbade dumping waste in the streets, as did the 1634 Lawes of the Markette, and similar orders were issued to the Worshipful Company of Fishmongers. More on this can be found in John Evelyn's 1661 treatise *Fumifugium (*also known by the less snappy but more amusing title of 'The inconveniencie of the aer and smoak of London dissipated together with some remedies humbly proposed by J.E. esq. to His Sacred Majestie, and to the Parliament now assembled').
Huge bonfires were also ordered to built at street intersections in an attempt to purify the air, and from various diaries anecdotal evidence lists people covering their faces with handkerchiefs and bundles of flowers (a posy) as they went about their business.
As for the practicalities of falling ill, Thomas Sherwood, a respected physician, in his 1641 work The Charitable Pestmaster details the process of infection.
16th century disease theory brought together history's three main ideas on the topic, religious punishment, miasma, and humoral imbalance.
Sinful living was believed to invite disease into the city from God, whereby rotting and putrefying materials (which were in no short supply) began to produce ill vapours, which then entered the body and imbalanced ones natural humours, leading to illness. Personal health determined how easily putrid air corrupted the body, and diet played a large role in this process. Not only was gluttony a sin and invited moral corruption, but it was believed excess food would putrefy in the stomach, creating a microcosm of miasma from within which threw off the body's ability to regulate the four humours as proposed by Hippocrates: blood, phlegm, black bile, and yellow bile. Physician Thomas Cogan proposed a hierarchy of humoral types of people, and how easily misama could imbalance their humours (in descending order):
So to conclude - yes, everyday people did believe in miasma, and I don't know of any sources from contemporary physicians questioning or denying the theory. Despite this though, miasma wasn't the only suspected cause of illness, and the variety of precautions against epidemics suggests that people were aware they didn't fully understand the process or mechanism of disease. Vermin and undomesticated animals were slaughtered en masse (in the case of plague, ironically worsening the problem) and widespread use of quarantine suggests people understood disease could be transmitted from person to person (though this isn't strictly true for plague).