Sometimes I raise them to level 3 and see if the first stat is 5 speed, which can probably improve one of my seldom used characters. Maybe level 6 if I have a lot of credits at the moment.
Going any further than that is probably not worth the risk for me. But that is coming from the perspective of someone who has been playing long enough to equip most of my characters with mods. I'm mainly looking for extremely good mods now, and if I get one then I will probably move an older mod down to a lower priority character.
For somebody who is starting from scratch and has many unmodded characters, it may be worthwhile to level up some white mods to 12, just to have something to equip even if it isn't ideal. If you level a mod to 12 and the stats are all bad, you can equip it on a pilot that you aren't using in squads, since ship stats are only affected by mod levels/dots and not by specific mod stats.