As found on the internet:
Belts and braces (a.k.a. bracers) are meant to hold one's trousers up. Going 'belt and braces' is a double insurance against having them fall down. The figurative use, as a general term for cautiousness, was coined around the mid-20th century.
In the USA 'belt and suspenders' is used with just the same meaning. That emerged at around the same time too. The first citation in print we have is from the Galveston Daily News, 1935 - in their 'News Behind the News' column:
"A pessimist wears both belt and suspenders."
GD&L
