How many centrioles/basal bodies are there in multi-ciliated cells throughout the cell cycle?
It seems that there is one basal body per cilium in multiciliated cells. Centriole duplication (as far as I understand, a basal body is just a different name for a centriole that is attached to a cilium) is tightly coupled to the cell-cycle. There is always one pair of centrioles that are at the base of the primary cilium (of which there is exactly one in most cells).

But not in multiciliated cells. These cells are differentiated and can no longer divide, and their centriole duplication process is decoupled from the cell cycle. In fact, they aren't, strictly speaking, "duplicating", since many procentriolar bodies may form around one cylinder. Indeed, each cilium has its own basal body; thus a cell with hundreds of cilia has hundreds of basal bodies.


Ishikawa 2011 is a review that examines the structure of primary and motile cilia, and ciliogenesis. It does not spell out how the basal body gets there.
Anderson 1971 describes the formation of new cilia in monkey oviduct. They first remove the ovaries of these monkeys to prevent cells from differentiating (this apparently requires estrogen), then administer estrogen and look at EM images of the cells over the following 6 days. They describe at length the processes of basal body formation.
Dirksen 1991 is another review focusing specifically on basal body formation during ciliogenesis. From their introduction:


  In order for the 200-300 cilia to form that the fully differentiated cell will need, production of that many centrioles is required first.
  The cell, after the last division, has usually retained a pair of
  centrioles. To what extent these mature centrioles are involved in the
  earlier stages of centriole formation, in which centriole precursor
  structures of varying sizes and shapes are generated and proceed to
  undergo transformational changes of great complexity, is largely
  unknown..

Sorokin 1968 is apparently a classic text which goes over EM images of ciliogenesis at various stages, and proposes a series of events describing what is happening. This is probably what you want, re: where are the other basal bodies coming from.
Bettencourt-Dias 2007 reviews the research on the formation of new basal bodies. This is another publication that answers your question very directly: 


  Many ciliated cells, such as those in vertebrate respiratory and
  reproductive tracts, can have 200–300 cilia per cell. This requires
  the generation of multiple centrioles during ciliogenesis. Here,
  centrioles are generated by two pathways, a centriolar and an
  acentriolar mechanism. In the first one, a parent centriole usually
  produces one daughter at a time; however, in certain cases several
  centrioles have been observed to develop simultaneously around one
  parent centriole, with the daughter centrioles being released into the
  cytoplasm to mature. The acentriolar pathway is the major pathway for
  basal body production. In this pathway, fibrous granules 124 that are
  70–100 nm in diameter first appear in the cytoplasm and subse quently
  move to the apical cytoplasm. Deuterosomes appear within the area.
  (...) Multiple procentrioles can grow out from deuterosomes, and
  mature daughter centrioles travel towards the apical region where they
  form ciliary basal bodies.