It seems likely that comedian Paula Poundstone originally brought the phrase to the national stage in the late 1980s and early 1990s, though it's the kind of thing that might have been spontaneously coined by parents multiple times throughout history (Poundstone's mother, at least, apparently said it before the comedian made the line famous).
Some early Poundstone citations are collected at BaryPopik.com, including an August 1989 Des Moines Register quote:
“She used to get mad over absolutely everything. I remember the time I knocked a Flintstones glass off the table and she said, ‘That’s why we can’t have nice things.’”.
Google Books offers the nearly-contemporary July/August issue of Mother Jones with a very similar story. Poundstone also apparently used this joke when she appeared on Comic Relief (I think in 1994), which may have given it further exposure. From there it likely percolated out into the collective consciousness where it was ripe for memic exploitation.
While I believe that Poundstone was the first to bring this joke to national attention, there is some indication that it was a gentle family joke before that. One piece of evidence comes from a memoir of the 1950s, published at about the same time Poundstone was beginning her standup career:
“Why don't you have nice things like Aunt Marion?” I asked my mother. “I have little children instead of nice things.” She smiled.
—Dorothy Allred Solomon, In My Father's House, 1984 (snippet view)
Evidence for the connection between breakage and "nice things" in motherly lectures goes back even further. From a 1905 "Lesson...for Little Children":
His mother stopped him, and said gently: “When you chased the cat and broke my beautiful vase, did I whip you?” Fred thought awhile. “No; you told how much you liked that vase, and said you were terribly sorry, and almost cried. You asked, ‘How do folks keep nice things?’ I felt very sorry. Now, I never play where nice things are, and don't break things.”
—Eliza Mowry Blven, "Lesson XXIII.—For Little Ones.", The Humanitarian Review, Volume 3, March, 1905
(I can imagine parents of small children lamenting the difficulty in keeping "nice things" whole since approximately the invention of pottery.)