There is no definitive explanation of the original decision to print U.S. currency in green ink. However, there are several reasons why currency is still printed with green ink today.
In 1929, when small denominations of banknotes were introduced, green pigment was easy to obtain in large quantities. The green ink was highly resistant to chemical changes, and the color has psychological associations with a strong, financially secure government.
Green ink also helps prevent counterfeiting. Counterfeiters discovered ways to circumvent early prevention efforts, by printing an imitation tint on top of counterfeit money. However, Tracy R. Edson developed a type of green ink that could not be removed from banknotes without also altering the black ink. Edson became a founder of the American Bank Note Company, which printed banknotes under contract with the U.S. government. His company used his green protective ink on these printed banknotes.
When the U.S. government transitioned from contracting outside printers to printing money entirely through the Bureau of Engraving and Printing, they continued using green ink to avoid changing the currency's appearance. There has not been a reason to stop using green ink or to switch to a different color, so it is still used for the sake of continuity.