Pew seems to indicate that the democratic party (or at least the electorate) has moved significantly leftward, particularly after the 2016 election. It doesn't seem that polarization is unique to a particular party. Moreover, I would argue that a self described socialist becoming one of the most important voices in the party is actually evidence of this leftward movement.
The whole point I'm making is that the Democratic Party has beat back the challenge from its base far more effectively than the Republicans, to the point that the argument just won't fit. Despite Sanders' high favorability among many Democratic voters, there's no evidence he's going to be very "influential" anytime soon.
They (The Democratic party) have been able to beat back challengers because their selection process for the presidential candidate is ironically less democratic than the Republicans. The GOP super delegates cannot vote against the way their state voted and they have less influence. The DNC super delegates have a greater influence and do not have to vote inline with their state.
As for Sanders. He has lost influence because he betrayed his base twice (he endorsed a candidate that the base consider to be totally corrupt) and lets face it he is too old now.
That's a factor but they have also been more willing to coordinate to stop outside challengers, like when everyone (including the frontrunner!) stepped aside to make way for Biden.
Base aside, I think Sanders isn't influential because the party insiders are strongly opposed to him. By way of contrast, despite a disastrous primary run, look at Kamala Harris.
[1] https://www.pewresearch.org/politics/interactives/political-...