What does mostly mean? I'm pretty sure (but can't prove) that every polynomial outputs infinitely many primes and infinitely many composites. I assume we're limiting to polynomials with integer coefficients.
By "mostly" we could possibly mean that the asymptotic density of one or the other is higher. For example the even numbers have asymptotic density in the naturals of 1/2. It's a way of quantifying some infinite sets where cardinality doesn't work.
The problem is that the primes have asymptotic density of zero. As numbers get large, the primes get further and further spread out. There are infinitely many of them, but there are huge runs of composites between them.
Questions like this are actually very deep as I understand it. I couldn't find any references that directly addressed your question but this article was interesting anyway.
Thank you for the knowledge, including the link. They may be the closest I will get to an answer. (By "mostly" I indeed mean a density greater than 1/2.)
So density might not be a test for the rate of finite polynomials generating primes or not, like dividing zero by zero. Is the answer different for transcendental polynomials, infinite or not, of irrational coefficients?
Here's another: does an asymptotic density of zero for primes mean that p(1)/1+p(2)/2+p(3)/3+p(4)/4+p(5)/5+...+p(N)/N=2/1+3/2+5/3+7/4+11/5+...+p(N)/N is finite, where N is an integer?