I have just done some renovations and will have to replace some quarter round. Attached is picture of existing. Is there a "proper" way to finish quarter round that is against wood floor? I have seen white (like the existing) and I have seen varnished to match the floor. The baseboard itself is to remain white.
I am one of those yahoos who prefer the shoe to match the floor color, although IMHO it is a matter of personal opinion. In the high end homes I help oversee being built in the Washington DC area, the color was determined by the architect or interior designer, of which many of these guys were highly regarded in their field and there was choices made of both types of finish. Again personal opinion...
In the case for the question, I would stick with the painted option, even though I prefer the stain. The stain would look different than the original, unless the floors will be refinished, then the floor finishers stain it along with the floor. The painted version will be easier to match up, and it may not hurt to "face off" the base with another coat to help hide the joint where the shoe and base meet. That way if the shoe is caulked in place, it has been done before, the caulk will receive a coat of paint to protect it.
Beyond personal color preference. One deciding point can be if you want the baseboard to look taller or shorter. In a house where the door & window trim is the same molding as the baseboard, it's more proper to color the quarter-round the same as the baseboard so everything is symmetrical.
In a house that has much taller or therefore wider baseboards than the door & window trims then the baseboard won't noticeably change if the quarter-round is hideously (sorry, I'm with personal privacy advocate on the coloring) matched to the flooring.
One final deciding factor is future re-painting or floor finishing. Paint is easily changed & affects nothing else. However, floor re-finishing to a different color or even just shade will mean the quarter-round needs to be addressed so it doesn't look frighteningly disgusting.