Pew Research has a posting on union membership and peoples attitudes about unions in the US.
It contained this graphic. In 201 total membership in public (Federal, State, and Local employees) AND private unions was 14.5 million people, roughly split 50-50 between public and private.
This is very different than the mix in 1983. Total union membership was 17.6 million (21% more than today), but in 1983 the private sector union membership was 68% of the total.
The public sector has increased membership by 26% (from 5.7 M to 7.2M) since 83' and private sector membership has decreased by 39% (from 11.9M to 7.3M).
About 33% of all public sector workers belong to a union. Only about 6.7% of hourly wage workers in the private sector in US belong to a union.