This is a bit of a tricky one really. What is your cousins motive - if there is a father out there unaware that he ever fathered a child it may seriously disrupt his life to suddenly find a child knocking on the door. On the other hand he may be delighted. How will your cousin feel if the birth father turns out to not want to know either?
Plus - the birth mother may say she knows who the birth father is, but no one can really be sure of that without a DNA test - how does he know that the birth mother is telling the truth (she has already covered up a major aspect of her life - your cousins birth - successfully for many years)?
From your cousins point of view I would write to the birth mother and explain as Tink outlines in a previous post.
Beyond information from the birth mother I dont really see how one could proceed (given birth father is not named on birth cert) without causing serious disruption to birth mothers life by asking questions and nosing about etc... which Im sure your cousin does not want to do.
Id also advise a little patience. I have a friend who contacted her birth mother and same scenario as your cousin, birth mother didnt want to meet her, no one in family knew about my friends birth etc (including birth father), but after a year or so of letters between birth mother and my friend (through agency) birth mother DID agree to meet, and although her own family (new husband, new children) still dont know anything, letters and photos have been exchanged at regular intervals since and it is beginning to look as though birth mother will in fact come clean with family one of these days and my friend will be 'allowed' to actually meet her half brothers and sisters etc.... The agency in this case advised patience to my friend and said in some cases birth mother thaws when her own family are older and wiser and its likely to cause less disruption in her marriage (kids grown up and able to understand situation, husband more mellow with age etc..).