Answering in terms of western Europe after 1200CE:
What did farmers use for boots?
Farmers are large field agriculturalists created by the British enclosures and revolutions, and by the French and liberal revolutions in Europe. Farmers didn't exist until very slightly before rubber shoes/boots, and were rich enough to afford leather shoes or boots and to avoid field work.1
I suspect you mean "what did peasants wear."
What did peasants wear?
In my judgement regarding peasants, all of the following would be used by all peasants, depending upon material availability, local culture, and wealth. In particular, it is my view that the poorest peasants (vagabonds, waste squatters, and cottars) would have habitually not worn shoes during feudal economic crises, war, or when the level of exploitation was high.2
Nothing (source: medieval hours books, in particular Tres belles heures du Duc de Berry, https://classconnection.s3.amazonaws.com/695/flashcards/977695/jpg/labors_of_the_months_in_tres_riches_heures_du_duc_de_berry11348516007253.jpg ).
Cloth or leather shoes, with or without foot bindings / foot wraps or stockings (Hours, again)
Sabots / Clogs (Wooden shoes)
What was, or were, the precursors to the current, modern-day rubber-soled heavy duty boot?
As a protective mud shoe, the sabot / clog.
Was there even as much a need to wear some heavy foot protection before the rise of modern industry anyway?
Some tools like the billhook have rounded ends to prevent damage on dropping or ground strike. Most tools required different skills and had different safety structures: compare the serpe or serpette or pruning knife to the secateurs.
These notes are a response to excellent comments on this answer.
- While I'd dispute whether free alloders or free tenants in chief or substantially free peasant owners or renters were farmers just because they had large fields, I would certainly agree that "farming" for market began in Europe in the Netherlands to my current knowledge and was a significant practice. I don't know enough about Dutch farm accounting in the late medieval to comment fully on class status or commodity production in terms of my theoretical apparatus. I understate the presence of farming in the Netherlands above for two reasons: it didn't "take off" like the English or French farming practices did; it distracts from my point that farming is a historically specific economic activity not a universal term for cultivation or agriculture.
- While the living standards of European peasants varied wildly, from freedom and plenty to dearth and slavery, peasant societies often reproduce poverty in a proportion of their community rapidly, this production of poverty being a result of the community's property behaviour without reference to exploitation by lords and priests. As an example, the late remnants of English feudalism in the form of commons involved commoners with quarter or less commons rights who were living as squat cottars and entirely dependent upon market relations (legal or illegal) for their subsistence. This kind of poverty has existed in European peasantries. This kind of poverty would result in barefootedness. I also believe that Hours books represent idealised visions of society based on the sumptuousness of the clothing depicted and would represent the broad scope of footwear, but not the rates of use.
Continued, in the matter of shoes: Dorothy Marshall (1926) The English Poor in the Eighteenth Century p113 adult shoes supplied on poor law 4/- stockings 1/6 per pair.(England 18th C) Shoes in pawn 1740. p97 late 17thC shows a rural parish child in a non-farming community going reshod year after year. p98 shows similar children habitually starving, I doubt they'd be shod.
Samantha Williams (2011) Poverty, Gender and Life-Cycle Under the English Poor Law, 1760-1834 p31 reconstructs the family life of Lucy Townsend and William Newman, both parish dependent on and off, Shefford a rural parish late 18th C. Newman was rateable for the poor law but in receipt of "considerable assistance" with shoes mentioned as considerable assistance.
These accounts are before the ramp-up of the poor law system with the early factory system, in one of the wealthiest societies of its era for the common person (England, 18th C), across the period, in the middle of the little ice age. And shoes are indicated as an item delivered under sufferance.
Being without shoes might mean early death in Europe in the early modern. Europe in the early modern was a set of societies that were quite willing to put people to death economically.