We do "more than 50%" quorum. Right now, with 5 players, that means we play if 3 players are available. (I'm the DM.)
Absent player PCs are still present if the game paused in the middle of action, or are busy doing something else (usually "nearby") if not. As a great "grey area" example, last session's game ended in the middle of a big fight with cultists in their lair. The bard's player couldn't make last night's game, so the bard hung back in cover using vicious mockery on the leaders until the fight concluded (the leaders fled, and three cultists surrendered). When the present players decided to pursue the cult leaders, the bard stayed behind to "watch the prisoners"; the rest of the party charged into the next room, where the boss battle happened. During that battle, the paladin was forced to retreat to the bard for healing, then return - taking him out of combat for a few rounds, but still allowing the party to benefit from the bard's spells somewhat.
If the party is in town or something, the absent PC is just busy with "other things", and I'll spend a little time with that player outside the session. If the party leaves town, the absent PC can be present "but has a cold" (explanation for why not participating as usual), or might be "I'll catch up with you later".
Our general rule, though, is the PC continues with the party, but uses only basic attacks or cantrips, so when the player can play, their resources are fully available. Unless the party all vote that more powerful action is needed - and I (DM) can veto. In a session a few months ago, the warlock player was absent when the traveling group was ambushed on the road. The warlock spent most of the fight using his favorite cantrips, but at a key moment when things were going against the party, the other players decided the warlock needed to cast his wall of fire. It seemed appropriate, everyone agreed, so the big spell was cast - and flipped the momentum in the party's favor at last! And then the players all whined "that was half my spell power!" as the absent player was wont to do when using one of his precious spell slots!
Generally, a present player takes "control" of an absent-player PC when necessary. When the only elven-speaking PC was an absent player, for example, one of the other players acted for him (under review and critique by the other players). [Some of my players seem to roll better when running others' PCs, so they tend to be given this job!]
[Many years past, we sent absent PCs to the "Land of Melon Balls". They were out of action, invulnerable, no matter what happened that night... but they were also absent from XP and treasure and such. (No explanation was given or required, although I think for one Planescape campaign we made this "planar instability" a part of the PCs' storyline!) The inconsistency bothered us as we got older, and this option fell by the wayside.]