On the contrary, a good capture card could increase the quality of your stream, potentially lighten the load on your main PC, and improve the performance of your games while streaming.
If you're asking on purely technical terms, however, then yes, they can. Capture cards often use Chroma Subsampling to reduce bandwidth requirements, which will reduce the final picture's quality. Though it's worth bearing in mind that once you upload your footage to a service like YouTube, they will severely drop the quality anyway. So there's not a tremendous loss overall as a result.