Paul Foot says that Shakespeare's sympathies were probably with Mark Antony, and against the conspirators, in Julius Caesar (March SR). This is usually the view that is taken, but it is also one that can be questioned.
There are good reasons to believe, when we look at the context in which Shakespeare wrote the play, that perhaps his attitude towards the conspirators was more complex. In a classic essay the Marxist historian Christopher Hill has examined the role of censorship in 17th century drama. He points out that Shakespeare switched to writing Roman plays from history in 1599, at a time when English history plays were being more and more heavily censored. This increased censorship was the response of the absolutist state at a time of growing political and economic crisis.
Many critics have identified a parallel between the ageing and decrepit figure of Julius Caesar and Queen Elizabeth I, then approaching the end of her long reign. Discussions about republican Rome were also at that time a way of masking criticisms of monarchy, in a society where open criticism was impossible. As Paul points out, although the play is ostensibly on the side of Caesar, Shakespeare also devotes a lot of time to a detailed portrayal of the hopes and fears of the conspirators. It could be argued that even to present Brutus and Cassius in this way was quite risky.
Just as Mark Antony defies the censorship imposed on him in his famous funeral oration, indirectly accusing the conspirators through his praise of Caesar, it is also possible that Shakespeare indirectly uses a play defending Caesar (and monarchy) to air certain views about political motivation and revolutionary action in a sympathetic way. It seems from Paul's review that this has been the line taken at the Barbican. Of course this is all speculation, but I see no reason why we should simply accept the establishment view of Shakespeare's ultra-conservative sympathies, when there is no real evidence for that either.