The theory behind a Boost is that it's something that's so integral to the character that it can't be depleted. Note the phrasing on Boost: "your pool increases by 1 after getting the information." If you have an investigation skill at 1, you can investigate for 1-cost clues all you want since your skill will always be returning to 1 after your search. At least in Ashen Stars it also means that people with a Boost can safely increase the cost of a core clue in a scene from 0 points 1 one with less fears about losing access their main investigation skill down the road.
Generally speaking though, since you can never run a boosted skill to zero, I wouldn't worry too much about running their Investigation skill down to zero. Instead, I'd give them opportunities to spend more points out of their Boosted skill in order to find out secondary things related to the mystery.