Red wines are usually pressed at a low Brix level.. This means a couple things. First, they're saturated with CO2, from the fermentation that's already taken place.. And also, they arent fermented dry yet, so they'll create more alcohol and CO2 still. This CO2 helps to act as a buffer, as the grapes are pressed but i believe (dont quote me here, havent made red wine from grapes yet) they also add a small amount of k-meta to the pressing to keep oxygen at bay.
With white wine grapes, the pressing is usually done before fermentation so oxidation is less of a fear, but k-meta is used during the pressing so that oxidation doesnt get out of control. The yeast will actually mitigate oxidation damage done prior to fermentation, by using up some of the oxygen within the must, during the ferment.
Either way, when the juice/must hits the bucket - you probably wouldnt want to take a very big whiff because it'll be mostly CO2 or K-meta, depending on what you're handling...
Wine at the pressing stage is a bit different than wines that have been stabilized and are working on clearing - with those, you most definitely want to limit the amount of splashing/oxygen exposure because oxidation is a real fear when wine is just bulk aging.
Leaving it in the bucket for any extended amount of time is a different story though. It ends up under airlock fairly quickly, generally.