According to the rulebook, you check at the beginning of your turn. If you fall below the threshold, you don't get to use the extra rig die.
Rulebook, Page 5, Setup step 4 wrote:
You will automatically unlock your second Rig die for use during any Round you begin with $10 or more, and your third Rig die on any Round you begin with $20 or more.
So, this is pretty clearly implying that, since you unlock it "during any round" where you have the cash, you do not get it when you don't have the cash anymore. If it were permanently unlocked, you'd expect something more like:
"You will permanently unlock your second rig die if you begin any round with $10 or more."
However, there are many rules in this game that I have interpreted it exactly opposite to the designer's wishes... so I guess we'll have to wait. I'm only saying this because of the word "unlock", which kinda makes you think it would be permanent. Unfortunately, there is no definition of the word "unlock".
Until then, I'd still recommend that you go with Rules As Written and only use your die if you had the cash that round.