  We study supersymmetric open strings in type IIB $AdS_3 \times S^3 \times S^3
\times S^1$ with mixed R-R and NS-NS fields. We focus on strings ending along a
straight line at the boundary of $AdS_3$, which can be interpreted as line
operators in a dual CFT$_2$. We study both classical configurations and
quadratic fluctuations around them. We find that strings sitting at a fixed
point in $S^3 \times S^3 \times S^1$, i.e. satisfying Dirichlet boundary
conditions, are 1/2 BPS. We also show that strings sitting at different points
of certain submanifolds of $S^3 \times S^3 \times S^1$ can still share some
fraction of the supersymmetry. This allows to define supersymmetric smeared
configurations by the superposition of them, which range from 1/2 BPS to 1/8
BPS. In addition to the smeared configurations, there are as well 1/4 BPS and
1/8 BPS strings satisfying Neumann boundary conditions. All these
supersymmetric strings are shown to be connected by a network of interpolating
BPS boundary conditions. Our study reveals the existence of a rich moduli of
supersymmetric open string configurations, for which the appearance of massless
fermionic fields in the spectrum of quadratic fluctuations is crucial.
