Sometimes you can get a sulfur or rotten egg odor in the primary. This happens more in a closed primary with an airlock or when a carboy is used for a primary container.

I recommend if you are having this problem go to an open container for the primary and stirring the primary once or twice a day during the primary fermentation. SO2 can build rapidly in the primary. Stirring can eliminate this problem. Also splash racking as you rack to the secondary.

By "open" you mean a bucket of some sort that is still covered by a cloth or something like that? Doesn't that run a higher risk of getting contaminated?

Yes, an open bucket. Covered with a towel. When the must is fermenting rapidly there is not much of a chance of having problems. The wine must be racked as it reaches 1.020 or so SG. Fermenting in a closed container can sometimes starve the yeast of oxygen. A sulphur odor can build quite rapidly.

I have never used an airlock on my wine primary. Although I am fairly new to beer making I also ferment my beer also in an open primary.

All that said, if I am using an open primary you can't go on a vacation for a couple weeks and not have some problems when you return. You must monitor it a couple of times a day.