The abortion explanation has been repeated so frequently that many people have accepted this explanation without digging further into the details. This is one of many explanations that Steven Pinker discusses in his book "Better Angels of our Nature." I am not sure it is as clear cut as you imagine: (I apologize for the lengthy excerpt)

When I told people I was writing a book on the historical decline of
violence, I was repeatedly informed that the phenomenon had already been
solved. Rates of violence have come down, they explained to me, because
after abortion was legalized by the 1973 Roe v. Wade U.S. Supreme Court
decision, the unwanted children who would ordinarily have grown up to be
criminals were not born in the first place, because their begrudging or
unfit mothers had had abortions instead. I first heard of this theory in
2001 when it was proposed by the economists John Donohue and Steven
Levitt, but it seemed too cute to be true.147 Any hypothesis that comes
out of left field to explain a massive social trend with a single
overlooked event will almost certainly turn out to be wrong, even if it
has some data supporting it at the time. But Levitt, together with the
journalist Stephen Dubner, popularized the theory in their bestseller
Freakonomics , and now a large proportion of the public believes that
crime went down in the 1990s because women aborted their crime-fated
fetuses in the 1970s.

To be fair, Levitt went on to argue that Roe v. Wade was just one of
four causes of the crime decline, and he has presented sophisticated
correlational statistics in support of the connection. For example, he
showed that the handful of states that legalized abortion before 1973
were the first to see their crime rates go down.148 But these statistics
compare the two ends of a long, hypothetical, and tenuous causal
chain—the availability of legal abortion as the first link and the
decline in crime two decades later as the last—and ignore all the links
in between. The links include the assumptions that legal abortion causes
fewer unwanted children, that unwanted children are more likely to become
criminals, and that the first abortion-culled generation was the one
spearheading the 1990s crime decline. But there are other explanations
for the overall correlation (for example, that the large liberal states
that first legalized abortion were also the first states to see the rise
and fall of the crack epidemic), and the intermediate links have turned
out to be fragile or nonexistent.149

To begin with, the freakonomics theory assumes that women were just as
likely to have conceived unwanted children before and after 1973, and
that the only difference was whether the children were born. But once
abortion was legalized, couples may have treated it as a backup method of
birth control and may have engaged in more unprotected sex. If the women
conceived more unwanted children in the first place, the option of
aborting more of them could leave the proportion of unwanted children the
same. In fact, the proportion of unwanted children could even have
increased if women were emboldened by the abortion option to have more
unprotected sex in the heat of the moment, but then procrastinated or had
second thoughts once they were pregnant. That may help explain why in the
years since 1973 the proportion of children born to women in the most
vulnerable categories—poor, single, teenage, and African American—did not
decrease, as the freakonomics theory would predict. It increased, and by
a lot.150

What about differences among individual women within a crime-prone
population? Here the freakonomics theory would seem to get things
backwards. Among women who are accidentally pregnant and unprepared to
raise a child, the ones who terminate their pregnancies are likely to be
forward-thinking, realistic, and disciplined, whereas the ones who carry
the child to term are more likely to be fatalistic, disorganized, or
immaturely focused on the thought of a cute baby rather than an unruly
adolescent. Several studies have borne this out.151 Young pregnant women
who opt for abortions get better grades, are less likely to be on
welfare, and are more likely to finish school than their counterparts who
have miscarriages or carry their pregnancies to term. The availability of
abortion thus may have led to a generation that is more prone to crime
because it weeded out just the children who, whether through genes or
environment, were most likely to exercise maturity and self-control.

Also, the freakonomists’ theory about the psychological causes of crime
comes right out of “Gee, Officer Krupke,” when a gang member says of his
parents, “They didn’t wanna have me, but somehow I was had. Leapin’
lizards! That’s why I’m so bad!” And it is about as plausible. Though
unwanted children may grow up to commit more crimes, it is more likely
that women in crime-prone environments have more unwanted children than
that unwantedness causes criminal behavior directly. In studies that pit
the effects of parenting against the effects of the children’s peer
environment, holding genes constant, the peer environment almost always
wins.152

Finally, if easy abortion after 1973 sculpted a more crime-averse
generation, the crime decline should have begun with the youngest
group and then crept up the age brackets as they got older. The
sixteen-year-olds of 1993, for example (who were born in 1977, when
abortions were in full swing), should have committed fewer crimes than
the sixteen-year-olds of 1983 (who were born in 1967, when abortion was
illegal). By similar logic, the twenty-two-year-olds of 1993 should have
remained violent, because they were born in pre-Roe 1971. Only in the
late 1990s, when the first post-Roe generation reached their twenties,
should the twenty-something age bracket have become less violent. In
fact, the opposite happened. When the first post-Roe generation came
of age in the late 1980s and early 1990s, they did not tug the
homicide statistics downward; they indulged in an unprecedented spree of
mayhem. The crime decline began when the older cohorts, born well before
Roe, laid down their guns and knives, and from them the lower homicide
rates trickled down the age scale.153