Finks are used at lower levels because it is more powerful compared to the towers. One major problem with finks is that it's housing is not good compared to the number of Booby Traps, so Finks are not the best against a base with them. Think of Finks as weaker versions of Fangs; they have higher damage than HP, and deal devestating amounts of it. Use them in numbers for maximum effect, beware of traps!
At higher levels, Finks are rarely used. The reason behind this is that it can only take 2-3 hits from a Cannon Tower of an average level, and can only take 1 hit from a Sniper tower. For housing space, you might as well use Pokeys. Use them only against farm bases. When Fink's claws are activated , many people use them to deactivate booby traps. Fink's slow speed make them very vunerrable to towers. However,Finks have one good use at later levels: they can be an alternative to Fangs if you're poor. They have the same trait: higher damage than health. Being said, Finks are fragile monsters, but are great for destroying the rest of a buildings after an attack. For example, your champion might have taken out all of the enemy's defences, and now you've only got a minute to destroy the rest of the buildings or you lose, so deploy some Finks to get rid of what's left of the enemy.
Finks are rarely used in the Monster Bunker because of their low health and very slow speed.