Well, then, let's get started.
https://www.nytimes.com/2009/02/03/health/03real.html : "In one study by Austrian researchers, for example, a group of 50 adults were split into two groups and tracked for six months. One group was instructed to use saunas regularly; the other group abstained. At the end of the study the sauna group had contracted fewer colds. “This was found particularly during the last three months of the study period, when the incidence was roughly halved compared to controls,” the scientists wrote. Other studies have found similar results. But doctors caution that saunas can be hazardous to those with heart or circulatory problems."
https://www.healthline.com/health/can-you-sweat-out-a-cold#s... : cites the same study and says it may prevent colds, but cites one study saying that hot dry air specifically (3-minute exposures thereto) doesn't help... seems not the regular kind of sauna: "Use of this type of sauna has little in common with that of a regular sauna, where one usually stays longer than 10 minutes and sweating is desired." Hmmph. I'll say that one is only partly relevant.
And the "Other studies have found similar results" link's abstract says this: "The common cold has great socio-economic impact. To date, no prophylaxis has been scientifically shown to be effective. A number of older reports implied that visits to the sauna and other thermotherapeutic measures might provide a certain degree of protection. More recent data suggest that this supposition is probably true. Nevertheless, there are a number of important questions still to be answered in this area."
Sounds like there is mild evidence in favor.