The idea of microscopic reversibility was born together with physical kinetics. In 1872, Ludwig Boltzmann represented kinetics of gases as statistical ensemble of elementary collisions. Equations of mechanics are reversible in time, hence, the reverse collisions obey the same laws. This reversibility of collisions is the first example of microreversibility. According to Boltzmann, this microreversibility implies the principle of detailed balance for collisions: at the equilibrium ensemble each collision is equilibrated by its reverse collision. These ideas of Boltzmann were analyzed in detail and generalized by Richard C. Tolman.
In chemistry, J. H. van't Hoff (1884) came up with the idea that equilibrium has dynamical nature and is a result of the balance between the forward and backward reaction rates. He did not study reaction mechanisms with many elementary reactions and could not formulate the principle of detailed balance for complex reactions. In 1901, Rudolf Wegscheider introduced the principle of detailed balance for complex chemical reactions. He found that for a complex reaction the principle of detailed balance implies important and non-trivial relations between reaction rate constants for different reactions. In particular, he demonstrated that the irreversible cycles of reaction are impossible and for the reversible cycles the product of constants of the forward reactions (in the "clockwise" direction) is equal to the product of constants of the reverse reactions (in the "anticlockwise" direction). Lars Onsager (1931) used these relations in his well-known work, without direct citation but with the following remark: 
The quantum theory of emission and absorption developed by Albert Einstein (1916, 1917) gives an example of application of the microreversibility and detailed balance to development of a new branch of kinetic theory.
Sometimes, the principle of detailed balance is formulated in the narrow sense, for chemical reactions only but in the history of physics it has the broader use: it was invented for collisions, used for emission and absorption of quanta, for transport processes and for many other phenomena.
In its modern form, the principle of microreversibility was published by Lewis (1925). In the classical textbooks full theory and many examples of applications are presented.