The key here is if you are talking mice or rats and how bad the infestation is.That will determine cost and extent of the problem.

It may attract tenants with good income and credit, but then you find out they throw a lot of parties, never change the air filter, never clean the dryer lint filter, have 4 dogs (especially once they see your big fenced-in back yard), and feed black mold infestations in the bathrooms because they don't have the good sense to turn the fan on or leave the door open after showering.There are certainly worse investments. 

I check for termites, mold, asbestos, infestations, and make sure the building is structurally sound.  

An undiscovered infestation of termites would be very bad. 

Examples: tenant has a filthy unit which leads to a perpetual infestation of roaches who take up residence in nearby units.

When it comes to water damage and especially mold, you don't want to take any chances due to health implications and the threat of spread/infestation.

Aside from the normal inspections (wood infestation, foundation, roof, etc) I also like to look at comparable rents in as small of a radius to be statistically significant to ensure when these tenants move out you'll be able to get about the same if not more rent. 

- Is it a massive infestation?

With my very first tenant from Hell, the property was left vermin infested with dead animals throughout the apartment.

A brief overview of the repairs include patching the old leaking roof multiple times in multiple locations, dealing with a rodent infestation in the ducts, flooding beneath the house and more.