One of my longest standing regulars doesn't have a mobile - nor a pc, or internet. He's great, punctual, timely and upon initial booking rang from his registered landline. He accepts if there is a problem after he's set off then we won't be able to contact each other. Because he's so reliable I'm happy giving out new addresses etc as he is really 'old school', you the type.
Anyway new clients these days are a different matter. They can book from any number belonging to them, so landline or mobile. If they use a payphone to book(quick google shows these up), then I won't guarantee the booking(I find payphone/withheld etc are notorious no showers). They can ring from a nearby payphone just before appointment time but if I've not had a number from them(mobile or land) at this point they will need to show me some ID before I'll let them in the house. I have another fairly long-term infrequent chap. Doesn't want to use his new mobile number, so has to accept he might get here and no-one be in if anything crops up. He's happy with that.
Another problem I've had with chaps using payphones is that ringing a mobile from them eats up any change they have - so if they don't quite hear the instructions properly, get lost, have to phone again this has caused problems and for both parties. So taking into account the above it's about common sense. No mobile then be prepared to prove who you are, and be straight forward.
Biggest problem larger than this, currently is the number of guys using their mobiles to request outcalls -then saying they don't have a landline to check, since they only need a mobile phone and get broadband either via SKY or mobile dongles.