This method of Crassus' (who is "Crassius," the bizarre forbidden love child of Crassus and Cassius? :P) is very poorly attested, appearing only in Plutarch. Plutarch does not in fact say that Crassus had a fire brigade at all, he says that he maintained five hundred δούλους ἀρχιτέκτονας καὶ οἰκοδόμους, "slaves who were architects and builders." Nowhere does Plutarch mention actually putting out fires, he says that he would buy up burning lots and those adjacent to them. The idea is that he could release the owners from the necessity of rebuilding the property--at high cost to themselves--or trying to find someone to buy it off them, a difficult proposition when a new owner would need to spend his own capital to develop and given the insecurity of the investment seeing as to how the lot was vacated. It's possible that Plutarch means that these five hundred builders acted as the final line of defense. At Rom, before the mass use of pressurized pumps (though Pliny seems to know of a similar contraption) the most common firefighting method was buckets of water or, when those were unavailable, rags soaked in vinegar or anything that could be used to smother a fire. Under Nero the provisions of buckets and vinegar were requirements in the city (one questions how strictly they were followed, given the continued nonchalance of landlords), but this was not so in Crassus' time. The last line of defense for a burning quarter of the city was to demolish buildings in the path of the fire--the jurists argued with each other over whether you could or could not sue your neighbor for knocking down your property if it was clear it would have been burned anyway. If, after all, the fire had no fuel it could not spread, and by creating a dead zone around it it could be contained. Possibly Plutarch means for these slaves to be serving this purpose, but it takes a generous reading of the text to make that work. Plutarch calls them τεχνίτας, from τέχνη, "skill, art"--a τεχνίτης is a skilled craftsman, an artisan, rather than manual labor of the type that would be knocking down houses. Moreover, Plutarch's comment is followed up by his assessment that despite owning so many skilled slave workers Crassus never built more than one house for himself, and goes on to list some of his other slave-worked properties. The emphasis seems quite clearly to be on the use of Crassus' staff as builders, not fire-fighters, and we must attribute the interpretation of the passage unambiguously as referring to fire-fighters to eager Victorian and early twentieth century commentators like Guglielmo Ferrero, whose work is quite amusing even in translation but which stinks of Whigishness through and through.
The threat of fire at Rome was well understood in the Republic, and steps to combat it began arising around Crassus' lifetime. Instruments of public order and safety were generally lacking under the Republic, in no small part due to the way that magistrates were often more privately funded from their own resources than publicly--in any case, this sort of thing was never understood as the primary purpose of the magistracies. The tresviri capitales (and their nocturnal counterparts, the quinqueviri cis Tiberim) were small commissions, probably employing a certain staff of public and private slaves, that at least in theory guarded against fire and theft. That they were ineffective is abundantly obvious, and they probably never had anything like the manpower or resources necessary to deal with great conflagrations. This sort of mass public provision from state funds, after all, was not really something the Republican state conceived of. Magistracies were unpaid, and magistrates expected to provide the capital necessary for their operations mainly from their own pockets--this was simultaneously a gesture of beneficence and a means of displaying one's dignitas on the grander scale requires by the intensely competitive nature of Roman society. The really important push towards the maintenance of a fire safety patrols operating from public funds came from M. Egnatius Rufus, who as aedile (26) maintained a large privately-funded gang of slaves to fight fires. His resulting popularity led him to seek the consulship, which Augustus refused--when his urban supporters protested he was executed for conspiracy against the state. As was so often the case, Augustus did not make the mistake of allowing a private citizen to override the constructed relationship between the populus Romanus and the princeps, which envisaged a tighter bond between the upper and lower levels of administration (see the reorganization of the vici and regiones and the establishment of freedmen cults in the vici). In 23 Augustus established a commission of 600 public slaves, granted to the aediles, for the purpose of putting out fires at the public expense. Soon after Augustus reorganized the vici he established that the individual vici had a duty to protect the local area from fire. By 6 AD he had established gangs of freedmen vigiles, which by the end of the Principate numbered some seven thousand, and which effectively tied the freedman population to Augustus as patron while maintaining a greater degree of public safety--provided, no less, not by outside agents but by the plebs themselves--than ever before. The vigiles, it should be noted, were not really fire-fighters, though. Besides their implicit role as political adjuncts to Augustus' persona, they were instruments of public order. Augustus harshly cracked down on the ability to display public dissent, even going so far as to limit the ability of individuals to meet in private, and the vigiles, who were a constant presence throughout all districts of the city, provided an excellent means to prevent dissent and assembly.