There is a cottage industry in the United States of manufacturing quotations and ascribing them to the American Founding Fathers. A recent one, "We are all born ignorant, but one must work hard to remain stupid," ascribed to Benjamin Franklin. The language and structure just don't sound like Franklin. The parsing of the difference between stupid and ignorant seems to have arisen after Franklin's death based on an N-gram comparison. Unfortunately, I don't have OED access to make that comparison in a more authoritative manner. The question is, does the quotation (as edited) sound as if it might have come out of the 18th century?
What Franklin didn't say
Like FumbleFingers, I can't find any attribution of the quoted language to Franklin before about the turn of the twenty-first century. (A Google Books search finds one instance from the year 2000, in 2001 Librarian's Engagement Calendar and Almanac.) On the other hand I did find the following quotation attributed to another popular source of folk wisdom—namely, "Chinese Proverb"—in Mary Cole, The Circles of Life (2008):
He who asks a question is a fool for five minutes; he who does not ask a question is a fool forever. We are all born ignorant, but to stay ignorant is a choice, without some form of education you are lost. —(Chinese Proverb)
And another folk favorite, Mark Twain, receives credit for this version of the saying, from Mohammad Mujeeb-ur-Rahman, The Betrayal of Intellect in Higher Education (1997):
Man is only born ignorant. It takes four years of college to make him stupid. —Mark Twain
The closest wording I've seen to the supposed Franklin quote before 2000 is from Earl Nightingale, This Is Earl Nightingale (1969) [combined snippets]:
Every human being has to be born ignorant and, for a time, live in ignorance. But if he remains ignorant that is his own fault. The fight against ignorance waged by everyone during his or her lifetime must be an individual, personal thing.
Unlike the other three authors I've cited so far, Nightingale takes full credit for his statement.
To me, the strongest argument against Franklin's responsibility for the cited quotation has two parts: (1) the quotation doesn't show up in the Google Books database until around 2000; and (2) when it does show up, no one cites the source in Franklin's work for it.
In the 520 Google Books search matches I looked at for the phrase "born ignorant," I found multiple references to (and some paraphrases of) relevant quotations from St. Augustine, Giordano Bruno, John Knox, Samuel von Pufendorf, Baruch (Benedict) Spinoza, John Locke, Thomas Halyburton, Isaac Watts, Claude Helvétius, Mary Wollstonecraft, Laurence Stern, Henry Ward Beecher, Thomas Hood, Edward Bulwer Lytton, Emmanuel Swedenborg, and Bertrand Russell—and all of them ultimately tied the quoted language to a particular work containing the phrase. The Franklin quote was the lone exception: No one offers any clear source in his work for it. I have no doubt that the attribution of this wording to Franklin is spurious.
What Franklin probably did say
Having rejected the attribution of the quotation cited by the OP to Franklin, I note that one source, Poor Richard's Almanac (1914), which consists of "Selections from the apothegms and proverbs" of Franklin, offers this otherwise unsourced saying:
Being ignorant is not so much a shame, as being unwilling to learn.
This quotation has a trail that goes back to Benjamin Franklin, Autobiography. Poor Richard. Letters. (1899), which lists it (with the word Shame capitalized) in a section titled "Sayings of Poor Richard, Selected from Dr. Benjamin Franklin's "Poor Richard's Almanack, 1733–1758." Even better, Bob Blaisdell, The Dover Anthology of American Literature, volume 1 (2014), assigns this Franklin quotation to the year 1755. That's the way a legitimate quotation tends to behave.
Where Franklin may have gotten the idea for what he said
A possible inspiration for the quotation (noted by Henry in his answer) is Claude Adrien Helvétius, De l'Esprit [On Mind] (1758). A translation of this book by W. Hooper, M.D., appeared in 1810 under the title A Treatise on Man; His Intellectual Faculties and His Education, with the following relevant passage:
OF FALSE SCIENCE, OR ACQUIRED IGNORANCE
Man is born ignorant ; he is not born a fool ; and it is not even without labour that he is made one. To be such, and to be able to extinguish in himself his natural lights, art and method must be used ; instruction must heap on him error upon error ; more he reads, the more numerous must be the prejudices he contracts.
A 1777 edition of Hooper's translation of the book replaces "fool" with "sot" and concludes the quoted paragraph with "he must have multiplied his prejudices by a multitude of lectures," but otherwise the versions of the paragraph are identical.
De l'Esprit may or may not be the source of the quotation attributed to Benjamin Franklin. It seems to be even more in tune with a quotation from Bertrand Russell (cited in FumbleFingers's answer): "Men are born ignorant, not stupid. They are made stupid by education." In fact, Russell's observation occurs—in his A History of Western Philosophy (1946)—in the context of a discussion of Helvétius:
Following Locke's doctrine that the mind is a tabula rasa, Helvetius considered the differences between individuals entirely due to differences of education: in every individual, his talents and his virtues are the effect of his instruction. Genius, he maintains, is often due to chance: if Shakespeare had not been caught poaching, he would have been a wool merchant. His interest in legislation comes from the doctrine that the principle instructors of adolescence are the forms of government and the consequent manners and customs. Men are born ignorant, not stupid. They are made stupid by education.
Helvétius in turn may have been influenced by Baruch Spinoza's Tractatus theologico-politicus (1670):
The natural right of every man therefore is determined by appetite and power, not by sound reason. For all are not constituted by nature to act according to the rules of reason. On the contrary, all are born ignorant of everything; and before they can know the true rule of life, and acquire virtuous habits, a great part of their lives must already have passed.
Benjamin Franklin was 52 when Helvétius published De l'Esprit, and it would be very surprising if he never read it, especially given that its "atheistic, utilitarian and egalitarian" ideas caused considerable scandal at the time it appeared (according to the Wikipedia article on Helvétius). Of course, if Franklin published his maxim about ignorance and shame in 1755 (as the Dover Anthology of American Literature asserts), he beat Helvétius to the punch by three years. In that case, Franklin's likeliest inspiration would have been John Locke, Of the Conduct of the Understanding (1706), who echoes Spinoza in his assessment of the human mind:
We are born ignorant of everything. The superficies of things that surround them make impressions on the negligent, but nobody penetrates into the inside without labour, attention, and industry.
If this is the correct chronology, then Franklin, not Helvétius, was the first to invert the argument of Spinoza and Locke that rising above one's native ignorance requires considerable effort (and presumably, education).
Given this citation from 1724...
To be Stupid and Ignorant is seldom the Character of a Thief.
Robberies on the High-way and other bold Crimes are generally perpetrated by Rogues of Spirit and a Genius, and Villains of any Fame are commonly subtle cunning Fellows...
...I think we can safely say OP's cited "Benjamin Franklin" quote would have been idiomatically perfectly credible in the 1700s. Whether it actually was from Franklin himself is another matter which I can't answer, but I will just note that all sorts of things get wrongly attributed to our favourite "epigrammists" such as Ben Franklin, Mark Twain, Winston Churchill, etc.
Having said that, I rather doubt Franklin did come up with this one. Firstly, it doesn't look to me like the type of thing he would say. Secondly, the earliest written instance of it I can find is 1999.
But mainly, I suspect it's a just mangled version of what Bertrand Russell (1872-1970) said...
Men are born ignorant, not stupid. They are made stupid by education.
Benjamin Franklin in famously xenophobic speech used after 1747, uses ignorant and stupid in the same sentence regarding German immigrants. Obviously a pattern he employed, not clear why the controversy.