Southwest already often isn't the cheapest, or even close to the cheapest, fare on many routes. And whether this causes an increase in fares will mostly depend on the existing load factor. If they already had planes mostly going out with a few empty seats, they don't necessarily have to change anything.
They are currently in the process of moving to a more flexible scheduling system (until now they could only have a weekday and weekend schedule) which should allow them to have more profitable flights (as they dont need to flt an empty plane for 3 days if only 2 days are high volume).
I've flown very often with Southwest, and have rarely seen a completely full plane. The fullest I've seen still has one or two middle seats free at the very back of the plane.
I can't even remember the last time I was on a flight (Southwest included) that wasn't at capacity. I'm sure it depends on where you're flying to and from though.
The time of day matters though. Early morning departures, in my experience - especially on weekends, are almost always partially empty. Last week I flew on a flight that left at 5:55 AM that was less than half full (25 people on a regional jet). Last December I flew on a southwest flight that left at 7:45 AM that was about half full (probably 60 people on a 737).
