Yes, they can be changed relatively easily by the client, simply by overwriting the current cookie with one of the same name, or editing it. What kind of verification is on your website? From what you are saying it sound's like it's just one thing, such as a username, which by changing that would give you access to another persons account?
You would be best off adding another cookie with the user's password hashed using sha1 or md5, or encrypted. That way the potential hacker would have to also figure out the user's password, but don't store the password as plain text.
Alternatively you may want to look at using sessions, I'm not sure about ASP but in PHP these store the data on the server, and can be accessed in a similar way to cookies.
Lee