Just how dangerous is ethidium bromide?
I usually find the treatment of Ethidium bromide in the lab widely exaggerates the true dangers of this chemical.
It is not really toxic (the LD50 is 1.5 g/kg bodyweight, as a comparision NaCl has a LD50 of 3 g/kg). It has also only been shown to exhibit mutagenic potential to bacteria after it had been treated with liver extracts. This means that some of the metabolized products of EtBr in the liver are mutagenic, but EtBr itself not. See reference 1 for details here. It is also not classified as a cancerogenic substance by the regulation bodies (for example see the National Toxicology Program of the US on Ethidium bromide). 

As already mentioned, it is used to cure Trypanosoma infections in cattle (reference 2). Here up to 1 mg/kg bodyweight are given, with no signs of toxic or other adverse effects. The typical concentration used in the lab to prepare gels is 1 μg/ml. It works as a drug by interfering with the mitochondrial DNA (changing it to the Z-form, which is not available for replication). See reference 3 for details. Ethidium bromide also acts as a Topoisomerase I inhibitor as other chemotherapeutic drugs. See reference 4 for more details.

Additionally, EtBr is negatively charged and cannot enter cells or the nucleus, since cannot pass the membranes. I would wear gloves and also apply the usual safety rules when working with gloves (do not touch anything you don't need to touch, especially things like door knobs, keyboards or phones as you might spread contamination) dispose your gloves when not needed anymore and you should be fine.



References:


Detection of carcinogens as mutagens in the Salmonella/microsome
test: assay of 300 chemicals.
Comparison of isometamidium chloride and homidium bromide as
prophylactic drugs for trypanosomiasis in cattle at Nguruman, Kenya.
A low dose of ethidium bromide leads to an increase of total
mitochondrial DNA while higher concentrations induce the mtDNA 4997
deletion in a human neuronal cell line.
The geometry of DNA supercoils modulates the DNA cleavage activity
of human topoisomerase I