Is it fair that a PhD student should be allowed to review a research paper?
Yes, it is fair.
- The quality of reviews written by highly experienced researchers is low.
- Many manuscripts have obvious flaws that should be spotted by the inexperienced.
- Reviewers are (in theory) supervised by the editor.
- Novice reviewers are more enthusiastic than experienced reviewers.
- If the paper cannot be understood by a PhD student, it will not be useful to very many people.
- PhD students may have more specialized expertise because they work on fewer projects, as compared to senior researchers (per @nayrb)
Of course, it depends on the individual PhD student. Some are better prepared to review than others.
In many disciplines of science, asking Ph.D. students to review papers is considered to be fair by the community. I can only speak for computer science here, but the following arguments may apply in other fields as well.
When a Ph.D. student reviews her/his first papers, this is typically done for his/her advisor. Advisors often check the reviews before turning them in, and thus avoid highly unfair or imprecise evaluation of the work of others.
Reviewing work is one of the best ways to get started with writing your own work. You get to know first hand what can make a paper weak and what you want to read as a reviewer. This is very valuable to the student.
A single review on its own is seldom the reason for acceptance/rejection of a paper. If there is a strong disagreement between the reviewers, then this is normally discussed in the case of conferences (which are prime publication venues in CS), or the editor has a look and will gather further information.
Ph.D. students often simply take more time to review a paper. Thus, they can find flaws in papers that senior researchers overlook in quicker reviews.
The assignment of referees to an article is not a random process where a computer draws a random number and matches it to "active researcher X" in the community. At least in my field the editors play a huge role in asking the "right people" to referee the article.
By making the focus on "inexperienced graduate students" you are focusing on the wrong things. An inexperienced graduate student who, nevertheless, has done research in similar fields and therefore has the expert knowledge to say something meaningful on the manuscript is certainly a much better referee than Prof. Dr. Messenberg whose recent research only overlaps with that of the manuscript in so much as that they use the same species of animal subjects.
The question you should ask is: "Should utterly unqualified persons be allowed to referee journal article submissions?" And the easy answer is, "No, since it is called peer-review not random-Joe-Schmoe-review."
You ask then: "How do we know whether someone is qualified?"
Answer: "You trust the editors; if you don't, don't send the manuscript to him/her/that journal."
As an aside: there are two ways that I've seen where Graduate Students come to review a paper.
The graduate student is passed the paper by his or her advisor to referee. Good advisors will only do so for appropriate papers, and may even provide guidance on how to referee a paper. Bad advisors are, well, bad (if an advisor gave an unsuitable paper to a student to referee, do you really think he himself will write a good referee report?)
The graduate student has independently came to the notice of the journal editor due to a previous interaction (conference presentation, paper submission, research discussion), and the journal editor feels that the student can fill the duties of the referee.