  We calculate the decay widths of the charmonium states, $J/\psi$, $\psi
(3686)$ and $\psi(3770)$, to $D\bar D$ pairs, as well as the decay width of
$D^* \rightarrow D\pi$, in isospin asymmetric strange hadronic matter, using a
field theoretical model for composite hadrons with quark constituents. For this
purpose we use the quark antiquark pair creation term of the free Dirac
Hamiltonian written in terms of the constituent quark field operators, and use
explicit charmonium, $D$, $\bar {D}$, $D^*$ and $\pi$ states to evaluate the
matrix elements for the charmonium as well as $D^*$ decay amplitudes. The
medium modifications of the partial decay widths of charmonium to $D\bar D$
pair, arising from the mass modifications of the $D(\bar D)$ and the charmonium
states calculated in a chiral effective model, are also included. The effects
of the isospin asymmetry, the strangeness fraction of the hadronic matter on
the masses of the charmonium states and $D(\bar D)$ mesons and hence on the
decay widths, have also been studied. The decay width of $D^* \rightarrow D
\pi$ in the hadronic matter has also been calculated within the composite quark
model in the present work, accounting for the medium modifications of the $D$
and $D^*$ masses. The density modifications of the charmonium states and
$D(D^*)$ mesons, which are observed to be appreciable at high densities, will
be of relevance in the compressed baryonic matter (CBM) experiments at the
future facility of FAIR, GSI, where charmed hadrons will be produced by
annihilation of antiprotons on nuclei. The interactions of the charmonium
states and $D(D^*)$ with the nuclear medium could lead to the possiblility of
the formation of exotic bound states of the nuclei with the (excited)
charmonium states as well as with $D(D^*)$ mesons.
