Word for solely being after a certificate or a degree?
It's more of a US expression rather than an international English one, but I think it fits; "paper chasing".
In order of increasing pejorative force, I offer you
certificate-chasing,
diploma-digging,
diploma-grubbing.
These all work with your example as they stand. With slight rephrasing,
mercenary
would work as well, although it is a more generally applicable term when standing alone, without context to define the domain of application.
To rephrase to fit your example, "Mercenary enrollees are a serious problem ...".