I have a finished basement with an office, guest bedroom and bathroom. All about 700 sq ft. I work 12+ hours in the office and really need to get some fresh air circulating by extracting stale air. I'm looking for a good method of doing this. I had considered running an air vent with a slow/continuations fan or a high volume fan that I can turn on a time for 15 minutes. The office is adjacent to a utility room and I'm concerned that too much of an exhaust could extinguish the water heater pilot light and cause other problem. I do have an air-in vents (heat/cool), but nothing going back out (circulating). The office is next to a window which opens to a window well. Any advice would be appreciated.
Energy Recovery Ventilation (ERV) You may want to consider either spot ERV or whole house ERV.
These devices both exhaust and intake fresh air, doing some energy exchange to reduce the impact on your air when the temperature differences between inside and outside are great.
As a spot unit, you could just run the unit in your office. The Panasonic unit I've read about has automatic controls, but I would supplement them with a countdown timer to limit use even further as necessary.
Leviton 12 Hour countdown timer
The divided wall cap means only 1 penetration is needed and it incorporates a pre filter for the incoming air.
Declaration: No financial interest in Panasonic or Leviton. I have used (personally) and installed (for customers) Panasonic fans and Leviton switches.
Making a few assumptions here inferred from the details in your questions so please comment to correct any mistakes in my assumptions. The basement is already vented from your A/C unit so you don't really want to open the window unless you have to. I'm guessing that the room does not have any ceiling fan or anything, that would help circulate the air in the room a bit but it could still be an extreme hot or cold area and feel stale. If the basement opens into the house and you don't mind leaving the door open, a simple solution would be a Door Frame Fan or Doorway Booster Fan (same thing different names) this mounts in the corner of the door frame and just helps circulate the air from room to room. A more full solution would be to install an exchange/return back to your A/C handler in the basement but that's probably not practical.
As for the pilot light concern, it would be near impossible to have an ambient flow in an adjacent room or hall to blow out a pilot light. A small amount of flow would actually provide more oxygen to the fire anyways. A large enough pull to create vacuum (in effect) in a large space would be nearly impossible as well as give you a headache and probably make you pass out before extinguishing a flame, a.k.a it's not a practical concern without an industrial fan in a small enclosed space.
If you have forced air for heating and cooling, a thermostat that would also run just the fan once in a while to provide circulation might be sufficient. Honeywell's VisonPro TH8000 is one model that has that feature. Look for "Circ" in the fan settings in the manual, which is available online at Honeywell's site here: http://yourhome.honeywell.com/home/products/thermostats/7-day-programmable/visionpro+8000.htm
You may find better options, but it seems to me that feature is the easiest remedy to your requirements. I would consider making sure that the door into this area allows enough airflow. Either with a sufficient gap at the bottom, or with a grate.