https://www.politifact.com/factchecks/2019/jun/21/donald-trump/donald-trump-again-falsely-says-obama-had-family-s/
Trump started publicly blaming Obama once he realized what a political loser this was.
However, Trump is the one who made "zero tolerance" family separation an official policy. And most Trump supporters backed him in this new ultra-hardline immigration stance, just as they support Trump in virtually everything. The explicit goal of this cruel policy was to deter border crossings.
The policy has since officially ended, but to this day no one knows for sure how many families remain separated.
https://en.wikipedia.org/wiki/Trump_administration_family_separation_policy
To wit:
"In January 2019, the administration acknowledged that thousands more children may have been separated from their families than the previously reported figure of 2,737, with officials uncertain of the exact number. Investigation showed that the child separation policy had actually begun in the summer of 2017, prior to the zero tolerance policy announced in April 2018. Federal officials said there were no plans to attempt to reunite these children because "it would destabilize the permanency of their existing home environment, and could be traumatic to the children. [30][31][32] In May 2019, the administration acknowledged that at least an additional 1,712 migrant children may have been separated from their parents even before the 'zero tolerance' policy was implemented.[33]"