I don't see how that should be a problem as long as you are wearing the proper protective clothing.
The hospital staff will be aware of the Hep C and will take all precautions themselves so surely they can protect you too.
If it's worrying you both, get your girlfriend to speak to her midwife or consultant for advice and clear up any other questions that she or you might have about the Hep C and child birth.
Hope this helps