Why does gaseous HCl not change dry blue litmus paper to red?
This is because gaseous HCl  does not dissociate to give up protons and chloride ions.
But, in aqueous solutions HCl does dissociate and exist in ionic form:

$$\ce{H2O(l) + HCl(aq) -&gt; H3O+ (aq) + Cl−(aq)}$$

Thus it acts as a typical Brønsted acid (proton donor) in an aqueous environment. In this case, water is gaining a proton $\ce{(H+)}$, so it is a base, while HCl is giving one away, so it is an acid.

Therefore it is able to turn the blue litmus paper red (a behaviour typical of acids).

For related topic please see: How does the litmus pH indicator work? as suggested above in comments.