Why aren't the amine groups in the nucleic acid bases protonated?
There are 2 answers:


As f' answered, the amino (NH2) group in a nucleobase is conjugated
with the heterocycle. Therefore, its pKa is closer to the one of
aniline. In other words, it is much less basic than an alkylamine.
Any drawing of a molecule in chemistry or biology is only a mere representation (agreed on by scientists) of the reality. For example, amino-acids are most usually drawn as H2N-R-COOH. We all know that it does not reflect reality but we all use that representation because it is much more readable than +H3N-R-COO- (which itself is also wrong), and because the real form of the amino acid will greatly depend on the experiment conditions. So you must always consider in what medium the molecule is to determine, more or less, its major form.