Wordorigins.org

The term booby meaning a stupid person dates to about 1599. From Patient Grissil, written 1599-1603:

Then, mage a pooby fool of Sir Owen. God’s plude, shall!

It probably comes from the Spanish bobo meaning the same thing as well as being the name of the type of bird—which are slow, stupid, and easy to kill. It is sometimes suggested that it comes from the German bube, which is sometimes used in the same sense, but the Lower German form, which would be closer to English, is boeve or boef, which makes the connection implausible.

The sense meaning breast is 20th century coinage. It is first cited from 1916 in Wentworth’s American Dialect Dictionary:

bubbies, n.pl. A woman’s breasts;—also boob(ie)s [...] 1916 cent. N.Y. boobies |bu’biz|. Common. Bubbies & boobs not known.

And from Henry Miller’s Tropic of Cancer, 1934:

She was lying on the divan with her boobies in her hands.

Although booby is a 20th century form, it traces back to the mid-17th century with the term bubby. The ultimate origin is uncertain, but it is likely related to the German bübbi, or teat.

(Sources: Oxford English Dictionary, 2nd Edition; Historical Dictionary of American Slang; Wentworth’s American Dialect Dictionary)