You may recall that the President of the United States is elected by an "Electoral College". Each state gets to choose their own electors. To this day, there is no requirement that there be an election involved. My understanding is (I wish I had a source on this) that in the first several elections, many states choose the electors through an act of the legislature. Also remember that most states had very restrictive franchise laws, limiting the vote to property-owning white men. So a turnout from those days will be difficult to compare to a modern election turnout.

Election turnout was largely based on the competitiveness of the elections within each state. For instance when Monroe ran for President in 1816 the turnout in Virginia was less than 10%. Virginia had only a few Federalist holdouts and the election was never in doubt. With a lack of competitiveness party organization tended to break down, and the machinery to turn out the vote simply wasn't in place or was "rusting". By contrast states like Delaware and the North East continued to have relatively high voter turnout well after Jefferson's election in 1800 for elections because the races remained competitive and both parties' organizations continued to exist on a state level.