1) Tap the Queen icon.
2) Tap one of the three specialists listed under "Current hire options"
3) Read the description and decide if this is the one you want. If not, tap "Go back" and repeat step 2).
4) If you found the specialist that will do the job (1/3 chance), tap "Hire now".
That's the hiring part.
5) Tap on the factory icon for Verne once. It will highlight and enlarge a bit.
6) Tap and hold on the factory and drag onto Vintgar. This will set up an empty sub into that direction, which will battle the 31 driller sub first. (The reason for this sub-sub battle, is that they're on the same path so they will meet each other.)
7) Your available specialists will be listed on the left of the popup. If you tap one, it will shift to the right side of the popup, displaying which drillers and specialists are sent off on the sub.
8) If you tap the correct one, the game will recognize your solution and this puzzle will be solved. If you tap the incorrect one, restart the game and change your pick in step 3-4.
There are only three possible hires, the Thief, Lieutenant and Saboteurs. Read their description to find out which one can stop the 31 driller sub from reaching Verne.