c. 1300, "breadth" (obsolete), from broad (adj.). Sense of "shallow, reedy lake formed by the expansion of a river over a flat surface" is a Norfolk dialect word from 1650s. Meaning "the broad part" of anything is by 1741.
Slang sense of "woman" is by 1911, perhaps suggestive of broad hips, but it also might trace to American English abroadwife, word for a woman (often a slave) away from her husband. Earliest use of the slang word suggests immorality or coarse, low-class women. Because of this negative association, and the rise of women's athletics, the track and field broad jump (1863) was changed to the long jump c. 1967.