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][1]
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  [1]: http://arxiv.org/abs/1306.2007