  Binary stars play a crucial role in our understanding of the formation and
evolution of star clusters and their stellar populations. We use Gaia Data
Release 3 to homogeneously analyze 78 Galactic open clusters and the unresolved
binary systems they host, each composed of two main sequence (MS) stars. We
first investigated the structural parameters of these clusters, such as the
core radius and the central density, and determined the cluster mass function
(MF) and total mass by interpolating the density profile of each cluster. We
measured the fraction of binaries with a large mass ratio and the fraction of
blue straggler stars (BSSs), and finally investigated possible connections
between the populations of binary stars and BSSs with the main parameters of
the host cluster. {Remarkably, we find that the MFs of 78 analyzed open
clusters follow a similar trend and are well reproduced by two single power-law
functions, with a change in slope around masses of 1$M_{\odot}$. The fraction
of binary stars ranges from $\sim$15\% to more than $\sim$60\% without
significant correlation with the mass and the age of the host cluster.
Moreover, we detect hints of a correlation between the total fraction of binary
stars and the central density of the host cluster. We compared the fraction of
binary stars with that of BSSs, finding that clusters with high and low central
density exhibit different trends. The fraction of binaries does not
significantly change with the mass of the primary star and the mass ratio. The
radial distribution of binary stars depends on cluster age. The binaries of
clusters younger than $\sim$800\,Myr typically show a flat radial distribution,
with some hints of a double peak. In contrast, the binaries of the remaining
clusters are more centrally concentrated than the single stars, which is
similar to what is observed in globular clusters.
