Let's differentiate between "cheap" and someone who doesn't have much of a budget.
I can work with someone that doesn't have much of a budget, but I can't help someone who is looking for the cheapest person to do the job.
A client that recognizes high quality work and knows the difference between poor design and great design with smart marketing is the person I want to work with. They understand that if they spend a little more, they will get a lot more in return. And they respect good work.
Before I went freelance over 25 years ago, I worked for a company that was always busy.... but the company folded and a few years later I saw my old boss, the owner of the company, selling computers at a big box store.
The reason his business failed is that it didn't matter what we did, we were always getting hammered on the price of our design services because my boss always sold "price" not service, quality, experience or results. The clients he attracted always asked for more and demanded to always spend less and less.
In this economy it's tough to say no to a prospective client, but it's important to remember that if you sell cheap you will attract cheap. And no matter what you charge a client that is cheap, they will always want to buy your services for even less!
At times I get prospective clients who tell me that they can get a project done for $50 by "this student I found on the internet" or "my cousin for $75" or "this company in India for $100". I can't help these people because they can't tell good work from bad work and are just looking for cheap. Plus the chances of their success and repeat business is low.
Bottom line: Work with someone who has a limited budget, if it's within reason. Run from the person who couldn't care less who they are working with and are just looking for cheap.