According to Suetonius they crossed a small bridge over the Rubicon. Specifically, he claims Caesar said "Once across that little bridge, we shall have to fight it out."

However, it should be kept in mind that Suetonius was writing in around 121 CE. This was a long time after the events had transpired and his description of the crossing also includes a divine apparition that urges the hesitant Caesar to cross.

In any case, from what I have read the Rubicon was small even in the days of Caesar. So it is entirely plausible that he may have been able to wade his legion through it.

Source: Suetonius, The Twelve Caesars