One of the many problems with the abortion-crime hypothesis is that it is US-specific. For example, in Romania the government banned abortion in 1966 and instituted harsh punishments for illegal abortions. Yet no one has shown that this correlated with a crime wave in the late 1980s as one would expect if free abortions really decreases crime.
There are other variables too. Soviet dictatorships were pretty safe, because they were totalitarian surveillance "fake it till you make it" "smile and nobody gets hurt" dystopias.
Theoretically the connection is sound (so I think it warrants a pretty solid prior probability), after all unwanted children lead to decrease in socioeconomic status, which is a very surefire way to have a lot of problems in life, and without a supportive family and state, the number of options quickly dwindles to illegal and not so great ones. (Basically the "not so great" option is that in the Soviet dictatorships around here - Eastern Europe - at least the full employment mandate meant that there was enough make believe work available to not starve. )
