Well, this is surely false for elliptic curves, i.e. when $g=1$.
What is true is that for any $t \in \mathbb{N}$ the number of elliptic curves in $A$ with $\deg _{\mathcal{L}} C \leq t$ is finite. However, it may happen that there exists elliptic curves on $A$ whose $\mathcal{L}$-degree is arbitrarily large.
This already happens in the case $A=E \times E$, where $E$ is an elliptic curve. For a discussion on this problem and bibliographical references you can look at the preprint by L. Guerra Elliptic curves of bounded degree in a polarized Abelian variety, http://arxiv.org/abs/1306.2007 .