In the 1970s, the Democratic Party's national platform had words of support for school choice -- the 1972 platform stated that Democrats would "channel financial aid by a Constitutional formula to children in non-public schools". After his election to the U.S. Senate in 1976, Daniel Patrick Moynihan (D-New York) authored a bill to provide tuition tax credits to families who send their children to private schools. Moynihan's bill did not become law, but it had roughly an equal number of Democrats and Republicans as co-sponsors.
Adam Emerson at the Thomas B. Forham Institute reminds us of this important slice of school choice history at the link below.
Emerson also notes that more and more Democrats have been voting for school choice programs in state legislatures.