I don't think she does cannonically, I know QUEENMASK on Deviantart has made masked manips of her using Diamond and Pearl screenshots. The only pokemon ladies that I know wore a surgical mask at least once were Misty in Whichever way the wind blows, Jessie in pretty much the first or second episode of the XY series, and Nurse Joy in the Sun/Moon series.
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RE: What Pokemon episodes Dawn uses Surgical Mask?