Does ethidium bromide stain RNA?

Yes, ethidium bromide stains RNA but with about 10 to 20 times lower efficiency to that of double-stranded DNA.

Ethidium bromide stains denatured RNA more easily. This is because the disruption of hydrogen bonding between nucleotides and the separation of previously annealed strands creates double-stranded regions that allow ethidium bromide to intercalate more easily.