From what I can tell, the International Brigades were mostly effective only for propaganda purposes and to camouflage the presence of Soviet assistance to the Republican government. The 32,000-35,000 men in the brigades were a grab bag of unemployed workers, middle class non-combatants, veterans from the first world war, etc; all motivated by a shared socialist ideology and anti-fascist outlook. Despite their generally noble intentions (a few were certainly adventure seekers), they were mostly unused to combat and lacked military training [2006, Beevor].
This was a problem given that one of the motivations of the International Brigades were to provide an example in military discipline and tactics to the equally unready Republican army. Soviet military advisors tried to provide adequate training to the brigades, but often ran into problems such as political squabbles between soviet and republican leaders, cultural differences, and varying quality of equipment (which was mostly poor). The International Brigades only lasted for about a year during which the battles in which they played a decisive roll appear to have been bloody stalemates or in the case of the Battle of Guadalajara, due more to the incompetence of the Italian allies of the nationalists.
The International Brigades, were driven in their formation and ultimate dissolution by larger regional factors. The Soviet Union was unwilling to provide too much open support for the Republican government for fear of straining relationships with Britain and France, which they felt they needed to maintain as an anti-fascist block against Italy and Germany. Britain and France were unwilling to involve themselves because the general governments of those countries were ambivalent of the fascist countries (in some sectors, most notably the navy and business, open supporters of the fascists) and were put off by the left-wing characteristics of the Republican government. America was unwilling to get involved in another major European war, but many of her prominent businessmen (such as Ford) openly supported the fascist rebellion.
The International Brigades were never able to achieve the same level of military effectiveness as the nationalist Army of Africa, and only appear to have made it more difficult for Republican Spain to gain the support of the non-fascist governments. It seems to me that despite the undoubted courage shown by the International Brigades, they faced the same insurmountable problems of the Republic at large; it is no wonder that they were not capable of changing what was ultimately a losing battle.