My wife actually did this for a while, not massively lucrative, but she loved doing it.
From my recollection there were certain things she had in place when offering the service, specific dog walkers insurance cover (PL insurance etc), key holders insurance (covering you for holding onto your clients house keys and if you lose them etc), various indemnity waivers, off the lead walk release form etc. (if interested I am sure she could dig out some copies for your reference).
Her attitude was she wanted to offer a service she would be happy to use with our dog, so she would have test walks with all new potential dogs and their owners, see how their dog mixed with others, only walked a maximum of three dogs at once, as she felt that was the limit for being in control if something happened. She would also offer morning and evening walks for those on early or lates, plus a letting out service for peoples who just wanted their let out into the garden at lunch.
She used to charge somewhere in the region of £15 an hour, so wasnt making a huge amount when you factor in insurance, petrol etc, but she did love it.
She stopped doing it about 3 years ago and still gets the odd call of ex customers asking if she could walk their dog.