Is there any way that a Jewish woman can be hired to work as a babysitter on Yom Tov if she lives outside of walking distance? What if we are not certain that she is Jewish (in this case she has a Jewish-sounding name but the family that employs her has never asked her explicitly and maybe its just her father who is Jewish etc.)? The family doesn’t know for sure she lives beyond walking distance. But they think she uses transit of some kind when she comes to them.
Given that it is Yom Tov, driving is only a derabanan even were she to do it herself. Getting on a subway or bus is probably mutar legamrei, mei’ikar hadin, assuming one doesn’t leave the techum (which I think we can assume is definitely the case unless it is an inter-city bus). The only issue is using the subway card, or the bus ticket, which is a form of mekach umemkar (technically, a monthly pass would not be so – it is proof that you have paid for the month, not paying something now for the trip, ואכמ”ל) and in the case of the subway card, also electricity.
Given that we don’t know if she is Jewish – as you point out, the only real indication is the last name, and that comes from the father – and that she might not even come by public transportation or need to, even if she chooses to do so – and given that the level of possible transgressions is quite low – I think there would be no problem having her come to babysit. This is even before factoring in the outside issues, like would she be doing melakhah if she stayed home, and the benefit of being with a religious family over Yom Tov. It is a larger conversation whether and how those might factor in.