Sure, consider an algorithm which so to speak generates all schemes and outputs those which are curves of fixed genus.

More precisely (but omitting most details):

We first need an algorithm which outputs all number fields (possibly with repetitions), in whatever format, say as the multiplication table on $\mathbf{Q}^d$. This is easy.

Then, for a fixed number field $E$ given as above, we can provide an algorithm which lists all closed subschemes (possibly with repetitions) of $\mathbf{P}^n$ ($n$ fixed), in the form of the defining equations. This is also easy because given a basis of $E$ over $\mathbf{Q}$ we have the height function on $E$.

Finally, algorithms in commutative algebra allow one to check whether a given subscheme of $\mathbf{P}^n$ is a curve of genus $g$. Since every curve embeds into $\mathbf{P}^3$, let's focus on this case. One can construct the minimal free resolution of the structure sheaf, which gives you the Hilbert polynomial, in particular the dimension, and the genus. Finally, it's also easy to check smoothness and connectedness.

Bringing it all together, we have an algorithm which runs forever and lists all pairs consisting of a number field $E$ and a set of equations of a genus $g$ curve in $\mathbf{P}^3_E$ (possibly with repetitions).