Actually I would say their employer has the responsibility to dig them out in that case.

I would not say it is MY responsibility to pay to dig them out though. Except indirectly as I purchase goods made from whatever they were digging for... which that money ends up going to their employer, which should be responsible for rescuing their employees.

Making the gov't pay for it just encourages employers to send people into mines they know are unsafe because if something happens they would not be the ones footing the bill.

For who could be free when every other man's humour might domineer over him? - John Locke (2nd Treatise, sect 57)