By definition, a basic graduate algebra course in a U.S. (or similar) university with a Ph.D. program in mathematics lasts part or all of an academic year and is taken by first (sometimes second) year graduate students who are usually attracted primarily to "pure" mathematics and hope for a career combining some mixture of teaching, problem-solving, basic research. This definition already covers a lot of possibilities, especially if broadened to include those institutions offering only a master's degree. Most users of MO have probably had (or avoided) such an algebra course along the way, beyond an undergraduate introduction.
There is a long "abstract" or "modern" algebra tradition going back to E. Noether and B. van der Waerden, but the steady growth of mathematics has added a huge amount of material to textbooks and has also created too much competition for the time of beginning graduate students. In practice many students can and do bypass algebra at this level. My own sporadic teaching of algebra took place in three quite different departments (Oregon, Courant, UMass) with varying research emphasis on algebraic number theory: the most likely place where mathematicians will really need a lot of Galois theory.
Galois theory has an illustrious history and (to quote Lang) "gives very quickly an impression of depth". It exposes students to real mathematics, combining the study of polynomial rings, fields, and groups in unexpected ways. But it also takes quite a bit of time to develop properly, together with supporting material. And people no longer care as much about solving polynomial equations exactly as about using sophisticated computational methods to estimate roots. In real life the eigenvalues of a big matrix are not estimated by factoring the characteristic polynomial.
Especially in a first semester algebra course taken by a wider range of students,
I've found it more rewarding to spend time developing the parallel between
finite abelian groups and finitely generated torsion modules over
in the theory of finitely generated modules over PIDs). This is challenging
material but gets at some of the canonical form theory for operators in the way
most mathematicians should understand it for theory and applications. The minimal polynomial comes into its own here.
Even in a second semester course, where tradition at UMass and many other departments has favored Galois theory, there may be a stronger case to make for teaching basic character theory of finite groups. This too is a meeting ground for many subjects and has even broader applicability than Galois theory when developed into full scale representation theory. (For number theorists, there is the neat proof that degrees of irreducible characters divide the group order.)
Working in algebraic Lie theory and representation theory, subjects unseen by most Ph.D. students, I am especially conscious of choices about which subjects students get exposed to formally. Algebraic and differential geometry often have their own standard (but not first year) courses in departments like UMass, but most people with a Ph.D. in mathematics get by without even those subjects in their background. "What should every mathematician know?" seems more elusive than ever.
Is Galois theory necessary (in a basic graduate algebra course)?
POSTSCRIPT: I appreciate the fact that so many people have actually given the whole issue careful thought, since it bothered me all through my own teaching years. With so little time and so much to learn, choices are inevitable. And it's always easiest to follow the existing course tradition and textbooks. My definition above of "basic graduate course" doesn't fit everywhere, to be sure, but U.S. students usually don't learn much mathematics before that level no matter what their potential is. So the issue won't go away in most U.S. universities that offer advanced work. (It will also continue to be true that most people with a Ph.D. here in "mathematical sciences" will never encounter rigorous Galois theory in courses or in real life.)