Are the determinantal ideals prime?

There are several ways to prove that $I$ is radical. By the way, the statement that $I$ is prime is equivalent to $I$ being radical and the zero set of $I$ being an irreducible algebraic set.

An approach using Gröbner bases can be found in Chapter 16 of Miller-Sturmfels, Combinatorial Commutative Algebra

An approach using sheaf cohomology can be found in Sections 6.1-6.2 of Weyman, Cohomology of Vector Bundles and Syzygies. This requires a lot more background knowledge.

There is also the approach using induction on the size of the matrix and localization arguments in Chapter 2 of Bruns-Vetter, Determinantal Rings. Link to book: http://www.home.uni-osnabrueck.de/wbruns/brunsw/detrings.pdf

This isn't really an answer but I can point you to a reference that might be of some help:

For a discussion of this example for the ideal generated by 2x2 minors of a 3x3 matrix, see this nice discussion in Eisenbud's Commutative Algebra, p 107.

It seems that the general case is much more difficult. Eisenbud also mentions that Bruns and Vetter, Determinantal Rings [1988] is a nice reference for the general case.

I hope someone else can come along to tell you something more useful!