A person may have a cold or any other disease that is an airborne-infection. Is their sperm also contagious? Should the person take any precautions?
I'm unsure about the use of the word "contagious" here. If you would describe it as an organism "catching something" from another only once, this answer applies (vaguely perhaps).
One example of "contagious" sperm: sea urchins spray their sperm (called milt) into the surrounding water, so that it can hopefully fertilize roe, which are fish eggs. The eggs are also expulsed for some species, but for others they stay on the surface of the female sea urchin, as this Wikipedia article explains.
Thus, it's entirely possible for a female sea urchin to be bombarded by free-floating sperm, to ultimately fertilize the eggs on its surface. I don't know if one would strictly consider this a contagion, but it's what I initially thought of after reading the question.
When referring to sexually transmitted disease however, sperm in itself is not contagious, as it would require the pathogen to "tag along with it" in whichever environment (whether it be airborne or aquatic). Also, if considering impregnation as "contagious", physical contact would certainly be required.
In short: For humans, no. For sea urchins (and likely other aquatic animals), sort of.