Like the rest of the paper money currently in circulation, the $20 bill (or $20 Federal Reserve Note as it should more properly be known) was redesigned in 1928 by Secretary to the Treasury Andrew W. Mellon when the old large notes were withdrawn and smaller notes introduced as a paper-saving exercise. The front (or 'face') of the small 20-dollar bill featured Andrew Jackson, while the back bore a vignette of the White House. As of 2010, this design has lasted with a few amendments until the present day. You can identify 20-dollar bills from the 1950s by paying attention to minor details.
Look for the words “In God We Trust.” Although widely used on coins, this motto did not become ubiquitous on paper currency until after a law was passed in 1955. If your 20-dollar bill doesn't bear these words, then it's either from the first half of the 1950s or an earlier decade.
Inspect the vignette of the White House on the back of the note. In the 1950s, this was adjusted to reflect the remodeling of this famous building enacted under President Truman, in particular the addition of the Truman Balcony. The absence of the balcony dates your 20 dollar bill to before the 1950s. To find out what the balcony looks like, visit the White House Museum website.
Check to see whether the note bears the words “Payable to the Bearer Upon Demand.” This formula harks from the time before the Treasury moved away from the gold standard, when any citizen could take a note to the U.S. Mint and redeem its value in gold coin. The phrase was dropped from paper currency in 1963 at the same time as the last remaining silver coinage was withdrawn. The presence of this phrase on your bill with date it to the early 1960s at the latest.
Unlike coins, which are issued by the U.S. Mint, paper money is the responsibility of the Bureau of Engraving and Printing.
Even a crumpled and hard-used modern 20 dollar bill should be distinguishable from a 1950s example thanks to the color-shifting ink that modern notes employ.
The average life-span of a 20-dollar bill is two years, so if you happen upon a purportedly 60-year-old bill looking fresh and crisp, it would be wise to be suspicious.