If it came down to those two? Pompey would crush Crassus like a bug. There’s a reason one was called Magnus and the other one got killed on his first campaign outside Italy.
I don't know if it would be that simple. It depends on when the civil war broke out and what the power dynamics in Rome were like leading up to that. Even though the fall of the Republic is the best documented period of Roman history, the way Crassus operated makes it more-or-less impossible to know how powerful he actually was (which was probably by design). But in this hypothetical "duumvirate" TL, Crassus would certainly have the backing of the optimates, who were disgusted by Pompey's rapid rise to power (he held the office of consul at an illegal young age and had held no other magistracy before). The fault lines would likely see the elder statesmen, patrician scions, and indebted property-owners side with Crassus while Pompey would have a following primarily from his comrades from the Civil war and Mithridatic war. This war would probably be seen as more-or-less a continuation of the Marius-Sulla civil war, with Crassus assuming control of the core of Sulla's coalition. Pompey meanwhile, would likely court the plebeian demagogues who were so active IOTL (Milo, Curio, Marc Antony, etc.) in order to disrupt Crassus' influence in the city. Not sure how the civil war would play out, but my money is on Crassus. He was a savvy operator and had distinguished military credentials, plus the backing of most of the senate; but then again, so did Pompey IOTL, so who's to say.
How Pompey operated anticipated Octavian's settlement (keep the forms but have a strongman giving directions from the wings) to a large extent.
Another important feature that is often overlooked is that when Pompey assumed the governorship of Spain after his second consulship, he never actually went to the province. This seems minor, but the constitutional settlements of Augustus were heavily based on this earlier precedent whereby a proconsul may govern a distant province through his appointed legates for a number of years.