In most museums I am familiar with, the staff and administration make
those decisions.  Often the board (or a committee thereof) has input in
the very early development stage of a specific exhibition, such as where
to get funding, or has the final approval rights over an entire
exhibition season (which may be a simple "rubber-stamp" procedure).

At the Art Center, the Director of Exhibitions (myself) has primary
responsibility for exhibition selection, development and execution but
as a matter of courtesy and to make the programs stronger, shares
information and progress with the Executive Director and the Associate
Director for Programs, and presents the season at the first meeting of
the Program Standing Committee of the Board.

Julia Moore
Indianapolis Art Center