  Droplets of absolutely stable strange quark matter (strangelets) immersed in
a lepton background may be the energetically preferred composition of strange
star crusts and of the interior of a new class of stars known as strangelet
dwarfs. In this work we calculate the surface tension $\sigma$ and the
curvature coefficient $\gamma$ of charged strangelets as a function of the
baryon number density, the temperature, the chemical potential of trapped
neutrinos, the strangelet size, the electric potential and the electric charge
at their boundary. Strange quark matter in chemical equilibrium and with global
electric charge neutrality is described within the MIT bag model. We focus on
three different astrophysical scenarios, namely cold strange stars, proto
strange stars and post merger strange stars. Finite size effects are
implemented within the multiple reflection expansion framework. We find that
$\sigma$ decreases significantly as the strangelet's boundary becomes more
positively charged. This occurs because $\sigma$ is dominated by the
contribution of $s$ quarks which are the most massive particles in the system.
Negatively charged $s$-quarks are suppressed in strangelets with a large
positive electric charge, diminishing their contribution to $\sigma$ and
resulting in smaller values of the total $\sigma$. We verify that the more
extreme astrophysical scenarios, with higher temperatures and higher neutrino
chemical potentials, allow higher positive values of the strangelet's electric
charge at the boundary and consequently smaller values of $\sigma$. In
contrast, $\gamma$ is strongly dominated by the density of light ($u$ and $d$)
quarks and is quite independent of the charge-per-baryon ratio, the temperature
and neutrino trapping. We discuss the relative importance of surface and
curvature effects as well as some astrophysical consequences of these results.
