I am a tenure track faculty member at a large university with unionized faculty. Being a graduate student can be quite similar to being a tenure track faculty member, so I thought I'd post a few comments.
Our union includes the adjuncts, so many of our graduate students are members of the union in that respect. We also have a closed shop, so everyone is automatically included in the union contract as soon as they are teaching. This avoids the awkwardness of choosing whether or not to join the union. There are also advantages of having faculty and students in the same union. When fulltimers negotiate too high a salary and the part-timers do their services for next-to-nothing, the university will just increasingly rely on part-time teaching.
The primary disadvantage I've seen so far with being unionized is that we are unionized across subjects and everyone is paid the same for their rank from computer science to history. This makes it very difficult to offer salaries attractive to computer scientists. This is a major disadvantage. Where I was a graduate student, the math fellows had a much better salary than the English fellows. One needs to make sure that, as members of a subject area which often receives better pay, we don't end up being paid less after unionization than before.
Also, our union has fairly expensive dues and I'm not sure I'd want to pay them if I were a graduate student. Certainly I wouldn't want to be paying dues to cover the cost of a lawyer every time a student wants to claim they were under duress during a qualifying exam.
That said, I think the biggest advantage of unionization is the ability to negotiate salary and benefits anonymously. Graduate students and tenure track faculty are not in a position to feel comfortable complaining about salary to the same people who can hold back their degree and tenure. Even if the advisor or chair would be supportive, most students would not dare speak up. With a closed shop union the members complain to the union which then complains to the university. The advisors and chair need not be involved. They would just be told how much it costs to have a graduate student and then they will hire or fire according to their budgets. Who is hired or fired would depend on the contract negotiation, but one would expect the department to maintain the right to fire on the basis of "no progress" or a failed qualifying exam.
Occasions when one might want to speak directly to the union would be to discuss issues like health insurance for a second adult (not a spouse) and childcare. There is no reason for someone to reveal they are gay or have children to their own department if the union can arrange things for them.
Of course, all the advantages I've mentioned are for being a member of a preexisting union. The act of unionizing itself is a difficult one which might be viewed as confrontational by some faculty. It would be interesting to see how the faculty in a department would vote on whether their students should be unionized.
Finally, one needs to question issues like research fellowships vs teaching fellowships. Researchers are working just as much as teachers and often longer hours. Is time spent doing research towards a thesis part of the job or a benefit? Our union recongnizes that research is part of our job and also part that many facult enjoy the most. It administers grants to release the faculty from teaching to get more time for research. It is important that the union the students join understands this dual nature of research time as both work and benefit. When the union does include research, how does that effect the NIH and the NSF and other grant awarding institutions which pay for most of the research fellowships?
