One thing to keep in mind is that if someone found you online, and is asking about your services online, they're at least considering entering into a business relationship with you online already...So you probably don't have as much convincing to do as you might think, though I totally understand why you're concerned about this.

Any way you do it, whether face to face or online, both the designer and customer are taking a leap of faith when they agree to deal with eachother. Not designing related, but look at eBay...there are millions of people every week sending money to merchants in good faith that they'll get what they bought. People are willing to take that leap.

Your clients can be convinced to have faith in you if you follow the good advice here as it all helps. The portfolio which shows not only what you can do, but what you've done. References are good. The contract Julian uses demonstrates professionalism all by itself, and that too will help convince a customer to go with you.

And Bas and dchelix are totally right as well regarding customers. If they're not willing to put down a deposit, you probably don't want to deal with them. Business relationships work both ways. The customer has to be professional too.