Every client (and yourself) should sanitise their hands before a treatment by washing them and using a nail brush under the free edge then Cool Blue before you begin ... This should be routine.
If your normal routine is to only use Cool Blue hand sanitiser, or something similar, without washing and you can clearly see that the client's nails are dirty ... Suggest she clean them well with a nail brush and anti-bac soap before you start. The nail brush should then go into a sanitising solution.
You do not have to say "yuck, your nails are gross, go and wash!" :biggrin:
A simple polite invitation to step over to the basin should not offend anyone ... If it is not routine then suggest it as if it were.
I never clean yuck from beneath a client's nails ... That is not my job. Filth under nails could contain blood, faeces, urine or any number of microbes. That is the client's responsibility.
Attending an appointment at a nail salon with dirt under the fingernails is like going to see the gynaecologist without making sure one is fresh and clean.