Any interstellar material would be redirected by the Heliopause in the Heliosphere (https://en.m.wikipedia.org/wiki/Heliosphere). The solar winds travel for a pretty long time and only get slowed down at the Heliopause (or mark the Heliopause by slowing down there, if you want to look at it that way). Besides creating a strong magnetic field, which deflects interstellar ions, the density of particle interaction would lead to the deflection of most uncharged interstellar atoms as well. If an interstellar cloud was able to get a detectable amount of matter towards Earth, it would be pretty bad news. I guess the upper atmosphere would be getting stripped away, continuing with lower parts of the atmosphere, then mountains depending on the density. Like using rougher and rougher abrasive paper on wood.
EDIT: Okay it seems like only large scale objects can surpass the Heliosphere. Interstellar dust gets electrically charged after passing through the Heliopause and deflected by the lorenz force. An insterstellar dust or gas cloud could never damage earths atmosphere due to this effect. (arxiv.org/ftp/arxiv/papers/1107/1107.0283.pdf Page 3, "Between the heliopause and termination shock,...")
(If large scale astroids are contained inside the interstellar gas/dust cloud they could, as mentioned, travel to earth and damage it, if they are considered part of the cloud.)