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