Leaks can be caused by a few things. One can be a dirty washer that seals to a seat within the faucet. This requires the faucet to be taken apart, and a new rubber or fiber washer installed along with a new seat installed. It is a DIY project, but you do need a few special tools to do this. They do not cost a lot, but unless your willing to take on this challange you may be better off with a plumber.
Also some newer faucets use a cartridge of sorts. This device allows for the water flow to come out of the faucet at different volumes and temps. These are typically just replaced. Again not a hard thing to do, but you do need to understand how to take the faucet apart.
Depending on the quality of the faucet you have will also depend if you want to spend the time and effort into fixing this or just put a new faucet in its place. There are several nice faucets for around $100 or so that are not to hard to change out and gives your kitchen a whole new look while stopping the leak all together.
To give you more ideas we would need to know what kind of faucet you have that is leaking. A photo of it also would be a great thing as well. Once we all know what you have, we can be a bit more detailed on what you should do to fix it.
That's a good idea. It's just a basic " home depot" cheap faucet, and is probably due to be replaced anyway!