Huh. I didn't know it was used as a dye, and in medicine?! That seems . . . interesting for something that you should wash your hands after handling. Although one of the cranky old scientists I work with likes to tell everyone that they used to inject people with ethidium bromide while he's bare-handing gels, to which I always reply that they used to bleed people too.
And to save people from having to look it up, we use ethidium bromide to mutagenize yeast and it is used in gels because it is a fluorescent DNA intercalator, so you look at it under UV and you can see where the DNA is on the gel. But uh . . . that means it sticks itself into DNA and is not picky about what DNA, so while it might not penetrate your skin it still seems like a really bad idea to handle it if you don't have to, especially if you have any cuts. Everyone else wears gloves.