Over a decade ago I heard a paper at the annual meeting of Bible scholars called the Evangelical Theological Society (ETS) on this question. The speaker said that everyone always capitalizes adjectives for the Quran and Talmud. Everyone writes Quranic or Talmudic, not quranic or talmudic.
We also routinely capitalize Midrashic, honoring Midrash.
We write Vedic, capitalized, as an adjective for the Hindu scriptures (e.g., Vedic Sanskrit).
But when it comes to the Bible, many Evangelicals write biblical, not Biblical. The ETS scholar suggested it makes no sense to honor the Talmud and Quran (and Mishna and Vedas) but not the Bible. Hence he suggested Biblical and Scriptural, not biblical and scriptural.
Of course, both are correct. And lower case predominates in Evangelical literature.
We at GES just made a policy that we capitalize Biblical and Scriptural as well as pronouns for deity (which typically are not capitalized in most Evangelical publications).
When I looked up Midrash at dictionary.com I found this explanation of Misrash: “an early Jewish interpretation of or commentary on a Biblical text clarifying or expounding a point or developing or illustrating a moral principle” (emphasis added). I was a bit surprised and pleased that dictionary.com capitalizes Biblical.
In our politically correct day and age, it is common to honor other religions and diminish or even dishonor Christianity. It is a little thing, but I think that ETS scholar was correct. Capitalizing Biblical and Scriptural honors God’s Word. That is something that we think is appropriate.
We are not, of course, criticizing the many who follow tradition and do not capitalize Biblical, Scriptural, and pronouns for deity. I know Evangelicals that do that are not disrespecting the Bible or God. But I hope more Evangelicals will begin to capitalize Biblical and Scriptural. After all, “The grass withers, the flower fades, but the Word of our God stands forever” (Isa 40:8).