= Solution
Regard $f,g$ as polynomials in $Y$ over $\mathbb C(X)$. Their coprimality and the preceding argument let Euclid's algorithm produce
$$
A(X,Y)f+B(X,Y)g=1
$$
with coefficients in $\mathbb C(X)$. Clearing denominators gives a nonzero $h(X)\in(f,g)$, so $I\cap\mathbb C[X]\ne0$. Interchanging $X,Y$ gives a nonzero $k(Y)\in I$.
In the quotient, $h(X)=k(Y)=0$. Reducing powers by these two univariate relations shows that the finitely many monomials
$$
X^iY^j,
\qquad 0\leq i<\deg h,
\quad0\leq j<\deg k,
$$
span. Therefore $\mathbb C[X,Y]/I$ is finite-dimensional.
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