I can't answer from a professional bass players point of view as I'm not one, but I can give experience as a professional studio sound engineer (in a previous life) who's had the luck to have worked with some of the best musicians (including Rhino and funnily enough Mikey Finn from T Rex!). And they all had one quality in common. Their ability to leave their egos at the studio door. Nobody wants to work with a premadonna.
As it's been mentioned above, the ability and confidence to get on with people is very important because walking into a session full of other musicians /producer /engineer etc is very daunting.
Being a professional is not the same as being professional. Turning up on time, having gear ready to go and sound good, knowing your chops and being able to play in the pocket. Learn a song structure quickly and listen to what the band/producer wants. Reading the dots wasn't always important (unless it was a complete band + orchestra setup) but knowing scales certainly was.
The need for a session musician also varied. A sax player for instance would be expected to solo well, a drummer be rock solid, singers to harmonise for backing vocals etc etc. Unfortunately, most bass players were used to turn a synth bass line into one with more feel. Fortunately, I never had to record a bass solo!
Plus luck and conections.
I thought the section in the Marcus Miller interview with Scot Devine (posted on here) where he talks about being a session player was very interesting.