Students are citizens and are the future stakeholders of any given society. It does not help a society to disregard their future stakeholders' grievances. Not allowing students to protest disregards their grievances and is not a good way to endear a society to those that it will depend on in the future. Therefore, students should not only be allowed to protest, they should be encouraged to protest.
Yes, students should be allowed to protest budget cuts. They are the ones responsible for keeping schools open, essentially. Schools are meant to be their start on a healthy and functional adult life. If the establishment is not being run correctly, or is experiencing detriments due to budget cuts, then students have every right to protest it.
Students need to take their education into their own hands, as it's clear that the ruling class has no interest in actually preparing for the future. Things have gotten bad enough in Brazil that students have woken up to this, and hopefully we reach a similar point in the United States, and other countries around the world do too.
Right now, it's more important for young people to get an education than it is for them to protest. Young people shouldn't be quickly learning how to be a part of being professional victims. There are enough professional victims in the world already. They do not pay taxes, so they should not complain about budgets.