Ignoring for the moment which side is right or wrong, or whether political beliefs are well-founded or ill-founded - which side do you think has more desire to win, or more desire to see its agenda enacted?
I would argue that, in general, liberals are slightly more fired-up than conservatives. One reason is that, on a number of issues, liberals have more to lose, or are more affected by, than conservatives. For instance, on the abortion issue, if abortion is upheld, that doesn’t affect pro-life people who want to keep their babies - abortion isn’t being imposed on people who don’t want abortions (at least not in America, for now.) But abortion being banned would cause a pregnant woman who wants an abortion to not be able to get one. Likewise for gay marriage - if gay marriage is upheld, it has little impact on a heterosexual couple, but the banning of SSM has major impact on a gay couple.
There are some issues on which the reverse is true (gun owners lean Republican, and banning guns would affect gun owners big time, while allowing gun ownership has less effect on people who don’t own guns, except perhaps an increase in the likelihood that they would be the victim of a shooting, which is still unlikely.) But by and large liberals are more motivated to win at the ballot box, and push for their side, than conservatives.