I wrote a referral scheme for a client recently. It makes sense, existing customers are a your target customers and they probably know other people like them who could be your new customers. A happy customer recommending your service to their friend, has a lot more weight, than you plugging your own service.
You do need a few customers to start with though.

What referrals are not

Its a not an affiliate scheme, although they are related an referral scheme is about a customer telling another customer about you. At worse an affiliate scheme is some random and potentially non ideal customer putting a link on their site to your site.
They aren’t saying I use these people they are really good, they could be for you, to someone they know. They are just broadcasting an advert with the aim of a financial reward. They probably are not using their knowledge of your service in order to aim at people it is suitable for.
So you may end up with a whole bunch of prospects who are less than ideal or a waste of time.

Do you need to financially incentivise people?

Often this is all thought about as if I give person X £1 for each person they refer then they will refer me lots of people who I can make £100 on. But I’m thinking you don’t buy good will. People want to tell other people how well they are doing and if your product works well to them they will tell their friends. Unless you do something really embarrassing for them (not sure how well referral would work for an STD clinic).
You can encourage your customers to give referrals in other ways as well. Send them flowers, thank them in person buy them bottles of wine, them referring you is personal, so hopefully the reward is personal.

Oh and hopefully deliver a fantastic product or service.