Ed, I talk a lot about finite verbs in  DOING GRAMMAR.  Essentially, I use
the idea of finiteness to differentiate verb phrases that can be predicates
from other types of verb phrases.  A clause has to have a noun phrase that
functions as a subject and a FINITE verb phrase, that is, a verb phrase in
which the first word in the main verb constituent shows either past or
present tense form (and agreement).
 
I find finiteness a useful concept.  Not only does it allow you to define
clauses, it also allows you to explain participial phrases, infinitive
phrases, and gerund phrases as nonfinite verb phrases.