The terminology in this area was established in the 1960s and 1970s, during the heyday of Transformational Grammar. At that time, the "base" denoted deep syntactic structure (D-structure), as opposed to surface syntactic structure (S-structure). The specific architecture of the syntax in the theory has long since been revamped entirely, but the terminology from that era remains.
Modern syntax still uses terminology such as "base generated" or "base position" when it wants to acknowledge that a given constituent is in a derived position, meaning that it has been moved or copied into its surface position. Modern GB (Government and Binding) and MP (Minimalist Program) theory do not, however, acknowledge deep structure in the sense that it was understood when "base component" was a clear reference to deep structure.
My understanding of deep structure is that it was not created by transformations, but rather the transformations applied to a deep structure yielding a surface structure. The lexicon provided the lexical items (words) for the construction of deep structure, but it played little to no role in the transformational genesis of surface structure (from deep structure).