But the group’s defense of the Charlottesville rally has crystallized a recurring challenge for the organization: how to pursue its First Amendment advocacy, even for hate-based groups, without alienating its supporters.

It has seen a backlash on social media after the violence in Charlottesville. Even Gov. Terry McAuliffe of Virginia asserted that because of the A.C.L.U.’s intervention, the rally was unwisely held downtown where it “became a powder keg.”

Ms. Sullivan acknowledged that the organization was aware of “a lot of threats” on social media of people saying they would drop their membership, although neither she nor the group’s executive director, Anthony D. Romero, would provide more details.

And with numerous alt-right rallies scheduled in the coming days and weeks, the A.C.L.U. also faces the question of how to respond to the next case involving a white nationalist rally that the local authorities try to block.

Another potential First Amendment showdown in Texas already looms. Citing safety concerns, Texas A&M on Monday canceled a white nationalist rally scheduled for Sept. 11. The rally organizer, Preston Wiginton, told the Texas Tribune that he might sue and he might seek the A.C.L.U.’s help.

Since Charlottesville, the A.C.L.U. has clearly wrestled with what to do when the next case comes along. It indicated on Thursday that, in evaluating whether to take free-speech cases involving public gatherings, the group would consider the potential for violence at the event and whether protesters were going to be carrying firearms. But Ms. Sullivan, the spokeswoman, said that was not a new policy; rather, it reflected existing practices.

“We make decisions on whom we’ll represent and in what context on a case-by-case basis,” the A.C.L.U. said in a statement on Thursday. “The horrible events in Charlottesville last weekend will certainly inform those decisions going forward.”