It depends on how you are listening to your music, as there are various methods I will list as many as I can think of and also how to do what you want:
1. If you are using the "Radio" feature:
This option will prevent that particular artist appearing again when the radio is automatically choosing song for you.
(Search for 'Nah, skip it' on that link (their anchor tags are a little messed up))
2. If you are playing from a your playlist
- Select your name in the top right and click "Playlists"
- Select whichever playlist it is
- Search for the song using the search box
- Right click it and 'Remove from Playlist'
- You can then use the little X in the corner of each song to remove them from your current queue
- The next time you import the playlist into the queue it will not contain those songs
Actually I think those are the only two ways to listen to music on Grooveshark so ignore my statement earlier about all the different ways you can play music.
I realise this isn't quite 'downvoting' the song, however if you are making use of the radio (which takes into account your last listened to music, liked songs etc this 'Nah, skip it' is essentially a down vote feature).