The following is an interesting look at the percentage point change in the employment-population ratio during the terms of the last twelve presidents. (Note that the terms of Kennedy, Johnson I, Nixon II, Ford, and Obama are not four year terms.)
Of note, the employment-population ratio increased during only three terms of Republican presidents (Nixon II1 and both four-year terms of Reagan) out of the nine terms for which it increased on a percentage point basis.
The last three terms (Bush W I, Bush W II and Obama) have seen deep decreases in the employment-population ratio. Both of these presidents were in office during recessions for which employment either never fully recovered or recovered over very long periods of time. (See the top graph here from Bill McBride at Calculated Risk.)
I don't typically put a whole lot of weight on the effect a president has on the macro economy, but this has some interesting revelations brings up some interesting questions.
1 - Nixon's second term lasted only 20 months, and if you combined it with Ford's 28 months in office, it would be negative.