Clothing is probably the least problematic thing. Every time some nurse or doctor uses a keyboard I want to scream. I regularly see doctors use the keyboard, wash their hands, and then use the keyboard again before beginning an examine. Recently I've started to see plastic covers on the keyboards, as if that mattered relative to the order of touching things.

Realistically, cross-contamination in the hospital is almost certainly far more dangerous and consequential. Hospitals are full of infectious people, afterall, and why MRSA is more common in hospitals than in other environments. I'd rather be treated by a garbage man fresh off his route who's more mindful about what he touches after washing his hands and after touching something possibly contaminated, even if his clothes are filthy.