This is not directly addressed in either Lucas's script or the novelization ghostwritten by Alan Dean Foster. However, my inference has always been that Leia is, as a Senator, a well known opponent of Imperial excesses of violence and tyranny. (The prologue of the novel does begin with a quote from Leia: "They were in the wrong place at the wrong time. Naturally they became heroes," which seems to be referring to the enemies of Imperial tyranny.)
The Empire is only about two decades old as the movie begins, and it is still not considered commonplace for a military vessel to attack a civilian one without provocation.* The citizens of the galaxy are still largely accustomed to the freedoms that they enjoyed under the Old Republic, but it is clear that incidents like Vader's capture of Leia's vessel are becoming an unpleasant reality; from Leia's, "Darth Vader. Only you could be so bold," it is also clear that Vader himself is a major instigator of such incidents.
The Senate is evidently not controlled by the Emperor. Planets like Alderaan are still able to select their own representatives, and many of those senators are evidently sympathetic to the Rebel Alliance. The rebels could naturally be seen as fighting against the excesses of the new Empire, and the duly elected representatives of the various planets would have a natural reasons to find common cause with the rebels. Even members of the Imperial Senate who may disagree with the rebels use of violence against the Imperial military may feel that the rebels's main grievances are valid.
Leia, coming from the peaceful planet of Alderaan (and a secret Rebel Alliance member), would be a natural voice against encroaching Imperial militarism and totalitarianism. Vader scoffs that the princess was "not on any mercy mission this time," which further suggests that she has a history of undertaking visits to planets or sectors that have suffered under the Empire. Attacking her (a sitting senator, a known opponent of Vader's faction in the Imperial government, an opponent of militarism, and someone known to be sympathetic to grievances against the Empire) and taking her captive would naturally tend to provoke an outcry from her Senate colleagues and generate sympathy for the anti-military causes she championed.
*Vader would disagree (correctly) that his attack on Leia's blockade runner was actually unprovoked. In the novel, he claims: You passed directly through a restricted system, ignoring numerous warnings and completely disregarding orders to turn about—until it no longer mattered.