Much of this is going to be dependent on how one defines "conservatism." If one is defining it as a demonstration of strength of central authority, then the Red Scare would have functioned as an example of the movement because it consolidated power in the governing body. The climate of fear and paranoia that was advocated in both Red Scares was representative of a conservative tendency to rest power with the central authority. If liberalism is seen as a movement that sought to empower individuals through a divided government or one where checks and balances is present, then the Red Scare could be seen as representative of its opposite. The consolidation of power into a centralized force where there was little in way of limitation of power from external sources is a trademark of both Red Scares. Individuals who fell victim to the Red Scares had little, if any, recourse for protection and experienced the full and complete force of centralized power. In this light, one could say that the Red Scares and the execution of governmental power in the wake of both helped to represent a potential downside to Conservatism.
The red scare after WWI was an example of conservatism because one of the major tenets of conservatism in the 20th century was anti-communism. Conservatives are strongly opposed to communism on a number of grounds.
Mos importantly, conservatives believe that the way things have been is generally good. They believe that the experience that societies gain over the years is going to be superior to any theory that someone thinks up in their own mind. Therefore, they would say, a capitalist and democratic system that has evoloved over centuries is better than a communist system that has never existed other than in the mind of political theorists.
Conservatives after WWI, then, did not want their society to be changed in radical ways. Therefore, they were worried about the possibility of radicals changing the system through violence. This is what led to the red scare and that is why the red scare was an example of conservatism.