Given a finite rational number $n$ such that $0<n<1$, how can one determine whether the sum:
$$ \sum_{b=2}^{\infty} n_b, $$
where $n_b$ denotes the $b$-ary interpretation of $n$, is rational or irrational? In other words, how can one determine the rationality of, for instance:
$$ 0.1234_2+0.1234_3+0.1234_4+0.1234_5+\ldots? $$
Your question statement is a little confusing -- how does one go from the number $n$ to the decimal expansion? I'll assume you just write it in base 10 and then move on. I will also assume that by finite you mean its decimal expansion is finite. So let $n_b = \sum_{i=1}^{n} a_i / b^i$, where $a_i$ is the $i$th digit from the decimal point.
If $a_1 \neq 0$, then the series does not converge because one has the sum $ \sum_{b=1}^{\infty} a_i/b$, which diverges because it is the harmonic series.
Otherwise, we have $a_1 = 0$, and the answer depends on an open question; in particular it depends on the positive integer values of $\zeta$. If only even terms appear, then it is definitely irrational because the sum is of the form $$ \sum_{i=1}^{n} c_i \pi^{2i} $$ and the powers of $\pi$ are linearly independent over $\mathbb{Q}$ ($\pi$ is transcendental). Otherwise, the answer depends on which values of the odd positive integers are irrational (not fully known, although there are infinitely many which are), and whether they are linearly independent to $\pi$.