I never expected to see the day when girls would get sunburned in the places they do today.

Will Rogers (1879-1935) American humorist

(Misattributed)

Widely attributed to Rogers, but I was unable to find it in any published primary source. That's because it appears to have been said by a different Will Rogers.

In The Pathfinder, "Art of Wisecracking Takes on New Significance," Issue 1866 (1929-10-05), the results of "Wisecrack Contest" among the weekly periodical's readers provides the following second place winner (earning it $10).

Grandpa Wayback rises to remark: “I never expected to live to see the day when the girls would get sunburned on the places they do now.” Won by Will B. Rogers, Atlanta, Ga

That is not the famous Oklahoman humorist (William Penn Adair Rogers), though the latter is mentioned (along with Ring Lardner) in the text of the story as a famous wisecracker.

This appears to be the origin of the quotation, and an explanation as to why it was quickly associated with the more famous figure by that name, an association that occurred very quickly when the Rogers from Georgia was forgotten.

Variants (mostly attributed to Rogers):

I never expected to see the day when the girls would get sunburned in the places they do now.
[Albert Shaw, ed., Review of Reviews (1935-02)]

I never expected to see the day when girls would get sunburned in the places they do.
[P.G. Wodehouse & Guy Bolton, Bring on the Girls: The Improbable Story of Our Life in Musical Comedy (1953)]

Few men expected to see the day when women would get sunburned in the places they do now.
[Louis T. Stanley, The London Season, "Feminine Wiles" (1956), used without attribution to Rogers]

I never expected to see the day when the girls would get sunburned in the places they do now.
[John Birch Society, American Opinion, Vol. 4 (1961)]