It's all depends, it's a secret recipe on how google algorithms even work and how they determine which site will be placed on first pages, of course back linking to relevant sites as well as keywords etc. does help, but I don't think this will dramatically hurt your seo.
That's a good idea, the drop-down won't be visible to guest but Google can still have access to the explore and you would have to login to get the drop-down. That way it shouldn't affect your seo. And if you would like to get rid of explore altogether you can just disable it.