Etymology
The derivation of "senescence" is clear: it comes from the Latin "senex", an old man. This is the root of the modern English "senescent", in which the suffix "-escent" has its common implication of a developing state ("becoming" or " beginning to assume, or in the process of assuming, a certain condition"). Other examples include adolescent, iridescent, tumescent and so on. Thus "senescent" means "growing old", "ageing", or "in the process of becoming old", and "senescence", the noun derived from it, simply means that process itself. All the major English dictionaries - Chambers, Collins and the Oxford English Dictionary (OED) define these words in near-identical terms.
Both "senescent" and "senescence" were appearing in print by the 17th century.
According to the OED's researchers, the first known use of "senescence" in print was in 1695, in John Woodward's "An Essay toward a Natural History of the Earth and Terrestrial Bodies, especially minerals, &c." Woodward reflected that it was reasonable to conclude that a divine power, having provided a planet perfectly suited to the needs of its human inhabitants, would "continue to preserve this Earth, to be a convenient Habitation for the future Races of Mankind" - it followed that "the Earth, Sea, and all natural things will continue in the state wherein they now are, without the least Senescence or Decay". Here, as is so often the case, senescence is coupled with a term for deterioration, dissolution or loss of potency, acquiring a similar meaning by association. However, a particularly interesting usage in the context of this Wiki is that of Bishop William Stubbs in Seventeen lectures on the study of mediaeval and modern history (1886), which is cited in the OED: "It is not a dead but a living language, senescent, perhaps, but in a green old age" (I have not yet been able to obtain the original article to discover which language he was describing). The concept of a "green old age" is somewhat different from the irreversible decline and decay which the language of senescence often conveys, and is more in keeping with the way at least some plant scientists regard the process.
The first uses of "senescence" in a genuinely biological context seem to have been during the 1870s; these early references were almost exclusively confined to the animal kingdom. Charles Sedgwick Minot was a particularly enthusiastic adopter of the term, and used it in the title of his 1891 article "Senescence and Rejuvenation" (Journal of Physiology 12: 97-192), though this may have been the first in an intended series under that title which was never completed, since it consists largely of careful and detailed measurements on weight gain in guinea-pigs. In his 1908 book "The Problem of Age, Growth, and Death. A Study of Cytomorphosis", Minot set forth his definition of the word: "With each successive generation of cells the power of growth diminishes... This loss of power I term senescence". In retrospect, it is perhaps unfortunate that plant biologists chose to adopt the same term to describe a phase of plant development which, while indeed characteristic of older leaves and other organs, has its own unique properties and does not necessarily correspond to an irreversible decline into elderly decrepitude, impotence and death.