As you say, the open case (for determining whether there are a finite or infinite number of rational points) is the case of genus $1$. (If you are restricting to smooth curves in the projective plane, then these are smooth plane cubics.)
This is expected to be decidable. Indeed, the procedure is known; what is currently unproved is that the procedure does what it is conjectured to do.
Firstly, one has to determine if the genus one curve has any rational point at all. One can first check this locally (i.e. over $\mathbb R$ and all $\mathbb Q_p$); if there are no points over one the completions of $\mathbb Q$, then
there are certainly no rational points. In practice, checking this is a matter of solving congruences and applying Hensel's lemma (and staring at a graph,
in the case of $\mathbb R$-points), and hence is effective.
If the curve has points over every completion, it lies in the Shafarevic--Tate group of its Jacobian, and determining whether it is the trivial element or not is equivalent to determining if the curve has at least one rational point. The Shafarevic --Tate group is conjectured to be finite, and if it is, it is effectively computable by infinite descent; so this should also be effective.
Finally, suppose that your genus one curve has a rational point. It is then an elliptic curve, and the Birch--Swinnterton-Dyer conjecture gives a (conjectural) criterion for whether it has finite or infinitely many points. One has to check whether the $L$-function of your elliptic curve is non-zero at $s=1$ or not. Determining whether a holomorphic function vanishes at a point or not is not necessarily effective in general, but in this case, it turns out (thanks to the modularity theorem of Wiles, and the method of modular symbols) that computing the value of the $L$-function at $s = 1$ is effective. (It becomes a combinatorial problem, rather than a true problem in analysis.)
Actually, assuming finiteness of the Shafarevic--Tate group, one can also solve the question of finite/infinite number of points by infinite descent as well; there is no need to use the connection with modular forms or the Birch--Swinnerton-Dyer conjecture. (I just have a fondness for the latter, which is why it came to mind first.)
In practice, the algorithm by infinite descent I've outlined (incorporating BSD or not, depending on your inclinations) should work for any given genus $1$ curve; if it doesn't, you will have found a counterexample to (at least) one of two conjectures that are both expected to be true: the Shafarevic--Tate conjecture and/or the Birch--Swinnerton-Dyer conjecture! I imagine that some form of this algorithm is implemented in modern computer algebra languages.