"Berkeley" refers to the University of California at Berkeley.
The term "co-ed" means a female college student studying at a college that accepts both men and women -- a "co-educational" college.
There was an era when most all colleges in the USA were either male-only or female-only. Then there was an era when previously male-only colleges opened themselves to "co-education" meaning that men and women could attend the same college and study together in the same classes. In this context, when a man refers to a "co-ed" he means a woman who is studying at the college alongside the men.
And "cat" in this context just means a man, not necessarily a man who is a jazz musician.
So the lyric means, in effect, "I'm going to be the kind of man that those shy female students at the University of California, Berkeley will find attractive."
The lyrics of this song are:
Gonna wear tight pants and grow me a cool Van Dyke.
Gonna be the kind of cat them bashful Berkeley co-eds like.
Trying to do my work, but the time just slips away.
‘Cause we live it up all night, and sit around and gripe all day.
Go right, go left, it's still the same dead end.
Better sing the Berkeley blues, there's Ronnie Reagan 'round the bend.
A "Van Dyke" is a style of closely-trimmed goatee that was popular among college men at the time.
The University of California, Berkeley was known from the 1960s through the 1970s for its culture of student protests against the Vietnam War, and other protest movements on other civil issues.
This song's lyrics cynically and unsympathetically reflect on the "blues" that the students who attend this university feel as they live college life while being at the center of a culture of protest.
The song ends with a reference to Ronald Reagan beginning his office as the Governor of California (his term was 1967-1975). Reagan was regarded as an "establishment" figure who disapproved of the culture of college protests. Of course he went on to become President of the United States of America (1981-1989).