From the Random House Historical Dictionary of American Slang (For each definition I've copied out, I've included the earliest citation.):

1.a. Gambling - a playing card, especially in three card monte. *1781 G. Parker, View of Society 2: Black-legs, who live by the broads* and the turf... *cant for cards.

b. a ticket of admission, transportation, etc. 1912 Field, Watch Yourself [referring to 1880's]: 'Fix the olly! I gave him broads to the show! He's right as guinea! Fix him! Have this cheap Green County bilk pinched! I'll land him in the quay!' All of this... meant that the owner wanted the winner of the capital prize arrested and thrown into jail.

2.a. a sexually promiscuous woman; a prostitute; slut. [Especially common in early use, and probably the original sense, now subsumed by (b)]. 1914 Jackson & Hellyer, Vocab. Crim. Slang: Broad, Noun Current among genteel grafters chiefly. A female confederate; a female companion, a woman of loose morals. Broad is derived from the far-fetched metaphore of "meal-ticket", signifying a female provider for a pimp, from the fanciful correspondence of a meal ticket to a railroad or other ticket.

b. a woman--often used contemptuously. 1911 Hampton's Mag. (Sept.): Pretty soon what was technically known as a "broad"--"broad" being the latest New Yorkese--hove into sight.

So, this dictionary thinks the general "woman" sense, comes from the "slut" sense. And the "Vocab. Crim. Slang" thinks that meaning is derived from the "ticket" sense. And I speculate (rather wildly) that the "ticket" meaning derives from the "card" sense.

Wendy