Computers are typically quite happy to live in "out of the way" places, as long as they have sufficient airflow. That's probably a good thing for two reasons. Fan noise is one, as you mentioned. Reduction in accident likelihood may be another.
Temperature: Yes. Temperature: The cooler, the better. (Not sure if that's absolutely always true, e.g. Antarctica, but in most habitable places in the world, you'll be better off with cool air.) So, basement is probably a good thing when it comes to temperature. Humidity: Sure, there are some ranges. Regarding humidity, whether the basement is a good thing, or not, probably depends on where you live.
Air Quality: Air quality is a good thing, sure. However, many computers do run in places of low air quality. Cleaning can prolong the life of computers, so they say. That is why an industry of cleaning supplies, namely canned compressed air, has been able to thrive. Airflow is certainly a good thing. Now I've seen quite a few computers (as a professional who worked for a company providing professional IT services to numerous other companies), and computers did typically last longer in the offices of lawyers or dentists, compared to mechanics. Why? Offices are kept clean. Some of the computers in places where people work would get extremely dusty, and it did seem to shorten life spans a bit.
If you have a table saw in your basement, the sawdust would be a bad thing, and so a filter may be more appropriate. Clean air is desirable. That said, filtering the air for anything but humidity is probably not necessary in many environments, as long as there is sufficient airflow and you keep things clean. If your air is clean, the circulation/fans ought to keep things clean enough that you don't need more filters.