I have found Neil R. Lightfoot's work about the Bible canon to be a helpful resource. (But for a more detailed account, see Bruce Metzger's work on the NT canon.) Lightfoot outlines seven reasons to "reject" the Deuterocanonicals (also known as the Apocrypha):
1. The books "were never" included in the OT canon.
2. Lightfoot also writes: "These books, as far as the evidence goes, were
never accepted as canonical by Jesus and his apostles." Neither the apostles
nor Jesus quoted from the Apocrypha.
3. No Jewish writer of the first century (such as Philo or Josephus) accepted these books as genuine. Jerome (translator of the Latin Vulgate) likewise believed that these books were "apocryphal."
4. The apocryphal writings contain historical, chronological, and geographical
errors. See Judith 1:1.
5. "There is no evidence that the [LXX] ever had a fixed or closed canon of
6. The books cannot be maintained "on a compromise basis." I.e. The Deuterocanonicals may possibly supplement (at times) or conversely be at odds with Scripture. Therefore, these works must not be accepted at all.
7. The "Roman Catholic Church" pronounced the OT Apocrypha (except 2 Esdras
and the Prayer of Manasseh) as "authoritative and canonical Scripture" at the Council of Trent. Yet conciliar authority is not a sufficient condition for determining a work to be canonical. Prior to Trent, Lightfoot states, there were officials of the "Roman Church" who spoke out against the canonicity of apocryphal works.
These points can be found on pp. 121-122 of Lightfoot's book, the Second edition which was published by the Baker Pub Group in August 1988. The work has since been updated.