The Roman emperors did in fact execute "philosophers" liberally before and after Musonius Rufus, so the emperors weren't afraid on 'bad PR' in terms of killing off philosophers specifically. During Musonius Rufus' lifetime, in the kill list are least Rubellius Plautus, Seneca, and Publius Clodius Thrasea Paetus under Nero (in AD 65-67); Helvidius Priscus Older under Vespasian (in c. AD 74); and Arulenus Rusticus, Herennius Senecio, and Helvidius Priscus Younger under Domitian (in c. AD 93). Whether a critical voice like Musonius Rufus was exiled, executed or spared was always much more about complicated social and political entanglements within the webs of Roman power than simply "philosophical criticism". This is going to be quite a long post, but all of this is rather complicated!
The reason I put "philosophers" in scare quotes is because, when thinking about what sort of role someone like Musonius Rufus played in Roman society, a Socrates-like philosopher who is primarily a teacher and thinker, criticising power structures from the outside, isn't a very accurate comparison. All these men were, above all, active statesmen and members of Roman elite, with all the social, cultural and political obligations and power that came with it. They did also teach, write or at least consume philosophy, and self-identified as Stoics, and used philosophical language to criticise authorities. Greek philosophy was, regardless of its complicated relationship with Roman power, incredibly popular. Huge numbers of Roman political elite had been "philosophers" in this sense already since at least the 1st c. BC.
In the 19th and 20th centuries, it was popular to talk of a period of "Stoic opposition" to the monarchy in the 1st c. AD, when a number of Stoics supposedly opposed monarchy on philosophical grounds, and were subsequently exiled or executed. However, Stoicism isn't inherently against monarchy (the same way e.g. socialism is), and the criticism of the these Stoic "philosophers" hurled against emperors was rarely altruistically motivated. The "Stoic opposition" was more about personal politics and antipathies, and attempts to regain the old powers and privileges of Roman ruling classes, by using philosophical language. Hence, the term "Stoic opposition" has fallen out of favour.
The Flavian emperors did issue big expulsion of "all" philosophers from Rome (and later, all of Italy) - Vespasian in AD 71/2 and Domitian in c. 90 and 93. These blanket expulsions, however, were more about taking advantage of the critics' self‐identification with philosophy. So, not actual opposition to philosophical practice per se. E.g. Dio Chrystosom was never affected by any hostilities against philosophers under Vespasian, he in fact flourished in Rome at this time. Dio was just as much a philosopher as his teacher Musonius Rufus, but he had enough slickness to brand himself more as a "sophist" or "rhetorician", and not to label himself as one of the "Stoic dissidents" - and he possibly wrote (philosophical) texts against Rufus and the other Stoics under Vespasian.
For Musonius Rufus specifically, we need to firstly appreciate the context. The 60's had been incredibly destabilising to the Principate. The disenchantment of the Senate over Nero's increasingly despotic acts, among them many controversial murders/executions, lead to the Pisonian conspiracy, the revolt of Vindex, Galba and the Praetorian Guard, and thus end of Nero's rein. This was immediately followed by the chaotic civil war, the Year of the Four Emperors (AD 69). Vespasian is the last one standing, but he's left with the incredibly difficult task to restore order, build authority for himself as the first non-Julio-Claudian emperor, and construct a working and stable relationship between the emperor and the Senate. One of his solutions was the expulsion of philosophers, in order to get rid of his most annoying critics.
Like Dio Chrystosom, Musonius Rufus wasn't initially affected by Vespasian's expulsion of philosophers - he was allowed to remain in Rome in 71/2. Why? John Penwill (2003) suggests a few different reasons. Musonius had already been forced out of Rome twice by Nero, in 60 and in 65, which maybe had something to do with his "critical" Stoic activities, but more so about his close association with a fellow Stoic Rubellius Plautus, Nero's cousin who, according to various rumours, was plotting to overthrow Nero as emperor. Nero executed Plautus, but Musonius was later pardoned by Galba. Maybe Vespasian simply did not want to appear to be repeating his predecessor's tyrannical mistakes. Musonius also spoke a lot about Stoic distain for luxury and wealth, and this sort of philosophy was useful to Vespasian; the Flavians actively tarnished the Julio-Claudians with similar charges of over-indulgence, in order to prop up their own image as the better, frugal and fiscal emperors. Musonius Rufus was also popular and well-connected, and although not a senator (he was equestrian), he prosecuted in important public trials and was otherwise active in the political sphere. Vespasian had to be very careful where to make friends and enemies in the Senate, and Musonius Rufus and his friends clearly weren't worth it.
However, Vespasian did eventually have to resort to exiling Musonius, too. Although Musounius had always been a loud and critical, sometimes tactless, voice in Roman politics, he had never really directly attacked emperors. Apparently this changed, perhaps simply because of some personal antipathies towards Vespasian. Musonius seems to have started writing poorly veiled direct insults at the emperor. E.g. in fragment 19 Rufus wrote that "a man who sleeps with a female slave is as morally degenerate as a woman who sleeps with a male slave" - controversial in Rome, as Romans considered having sex with/raping female slaves very normal masculine business, but citizen women were strictly (by Vespasian's own law) expected to sleep with only their citizen husbands. This was a direct and personal attack against Vespasian, who, as was publicly known, indulged in sex with female slaves, and the great love of his life (but not wife) had been Caenis, a former slave. Clearly Vespasian at some point grew tired of Musonius' antics and exiled him, perhaps around AD 75. It is possible that Musonius also had strong connections with the fellow Stoic, and much more radically critical, Helvidius Priscus, so when Priscus was exiled, Musonius had to go too.
But, note that we know that Musonius returned to Rome after Vespasian's death, and presumably Domitian never had any problem with Musonius or his philosophy, since we never hear of him in relation to the many Stoic purges that Domitian did in Rome. [Though, it is possible that Musonius died already under Titus; we don't know when he died]. So, there clearly were much more bigger thorns in emperors' very sore sides than Musonius. And, it really was not in the interest of Vespasian to mindlessly execute critics; exile was just as good a method to silence someone. Vespasian was keen to demonstrate that he was not a mindless Julio-Claudian tyrant like Nero, executing people left and right. Showing mercy to high-born dissidents and criminals was more subtle and, usually much more effective, method for the emperor to control and sway the Senate.
Helvidius Priscus, however, was not as lucky. Also spared from the first expulsion of philosophers, Helvidius was later exiled and even executed. Why execute Helvidius but not Musonius?
It is clear that Helvidius' actions agains Vespasian were of a much more dangerous and persistent kind, and - it must be said - Vespasian did put up with Helvidius for a long time before he finally took action. Helvidius had enjoyed a much more illustrious political and military career than Musonius, high offices and successful campaign in Armenia etc. Helvidius was through his status, experience and connections more powerful and dangerous, but also more difficult to remove than Musonius. Helvidius' antipathies towards Vespasian were probably of a more personal kind than simply any "Stoic opposition" to monarchy. Upon his accession, Vespasian put an end to Helvidius' repeated attempts to prosecute senators that (he believed) had wronged his father-in-law under Nero's shenanigans, which might have been one of the reasons Helvidius grew disillusioned and enraged. Helvidius apparently, from day one, very publicly refused to acknowledge Vespasian as emperor, even in his office as prateor in AD 70. Helvidius also unsuccessfully proposed laws that would have limited emperor's powers, such as returning the management of public treasury to the Senate. Apparently Helvidius' insults towards Vespasian grew more and more strident over time, and presumably inaction by Vespasian was starting to compromise his credibility. According to Suetonius, Vespasian was reluctant to execute Helvidius to the very end:
It cannot readily be shown that any innocent person was punished save in Vespasian's absence and without his knowledge, or at any rate against his will and by misleading him. Although Helvidius Priscus was the only one who greeted him on his return from Syria by his private name of "Vespasian," and moreover in his praetorship left the emperor unhonoured and unmentioned in all his edicts. He did not show anger until by the extravagance of his railing Helvidius had all but degraded him. But even in his case, though he did banish him and later order his death, he was most anxious for any means of saving him, and sent messengers to recall those who were to slay him; and he would have saved him, but for a false report that Helvidius had already been done to death. Certainly he never took pleasure in the death of anyone, but even wept and sighed over those who suffered merited punishment.
Suet. 15