Finding someone you feel comfortable leaving your kids with can be a challenge, so often the best way is to go off personal recommendations. If no one you know can do it, try a professional "matching" agency like Findababysitter (Find a Babysitter).
Don't feel uncomfortable asking for references – it's always a good idea to give them a call so you know about the kind of prior experience they've had.
And lastly you may want to interview them. Ask some hypotheticals – i.e. "who would you call if this happened?" Or "what would you do if…" to help put your mind at ease.