I do both, sometimes shooting wide open, other times stopping the lens down a bit... Among other things, it depends upon the look I'm going for, the distance between me and the the subject, and how much separation there is between the subject and background. That's why they gave us all those f-stops to work with, after all! I generally don't go much smaller than the diffraction limited aperture for the camera I'm using, unless I'm seriously needing some more DOF, such as with marcro or landscape shots.Speed Barrels, CSHA Region 3 Gymkhana, May 15, 2010. EF 300mm f2.8 IS lens at f5.6. EOS 7D at ISO 200, 1/1000 shutter speed. Gitzo 1325 tripod, Kirk BH-1 ballhead, Wimberley Sidekick, ambient light.
Some equestrian examples....
In the image below I used 300/2.8 IS at f5.6... horse's face is tack sharp, rider is slightly behind the ideal plane of focus and a wee bit soft. What little background there is in the image is far away and more than adequately blurred down. With a fast moving subject up close (coming toward me at a full gallop), stopping down a bit also can help reduce focus error.
Other times, I want a bigger aperture to be sure to place the emphasis in one area of the image, soften the less important areas... Here I wanted the rider blurred more heavily and the emphasis on the horse's face. The difference is more obvious in big prints, than at Internet resolutions...Sadie & Karen, after the dance...Hossmoor Dressage Show, October 27, 2007.EF 300mm f2.8 lens at f2.8. EOS 30D at ISO 640, 1/500 shutter speed. Gitzo 1325 tripod, Kirk BH-1 ballhead, Wimberley Sidekick, ambient light.
Plus in the above image it's a slower moving subject so easier to maintain precise focus on. It's also indoors and using a camera that's a little more limited on high ISO performance, so a bigger aperture came in handy in this case for several reasons.
Other lenses improve nicely stopped down slightly...Cloverleaf, CSHA Region 3 gymkhana, April 23, 2011EF 300mm f4 IS lens at f5.6. EOS 7D at ISO 400, 1/1250 shutter speed. Handheld, ambient light.
The 300/4 IS is very slightly soft wide open. Images are very usable, but there's just enough difference I can tell the f4 lens from the f2.8 lens, when used wide open. The f2.8 lens is clearly optimized for wide open use... The f4 lens not quite. On the other hand, with both lenses at f5.6 it's nearly impossible to tell the 300/4 lens from it's 300/2.8 big brother.
Finally, stopping down a little can help cover up "user errors", for example...Sisco & Lucinda, ACTHA Ride for the Mustangs, 2011EF 300/4 IS lens at f7.1, EOS 7D at ISO 400, 1/2000 shutter speed. Handheld, ambient light.
Shooting fast to catch the horses with ears alert, the focus locked on the weeds a few feet in front of horses & riders... but thanks to using a smaller aperture and additional depth of field the image is usable, hard to tell at Internet resolutions but this image, which is also cropped a bit, actually prints quite good up about perhaps 8x10.
A football game, shooting the players across the field in front of the stands, I might use a really large aperture, too... to help the subject(s) stand out against the busy background of the spectators. Shooting down the field, OTOH, I might use smaller apertures for more depth of field... depending upon the background.... such as to keep the goal posts from blurring away to nothing.
I guess I just don't see any reason not to try different things, to go for different effects.