Are you a Tarantula hobbyist? If so, we invite you to join our community! Once you join you'll be able to post messages, upload pictures of your pets and enclosures and chat with other Tarantula enthusiasts. Sign up today!
To me, damp, plants and moss is fine but that fusty/musty smell you describe translates to mould, which flourishes in the presence of moisture and inadequate ventilation. A lot of keepers say mould is harmless and in the main it is, you can just scoop it out, and get cleaner insects like Panzo says to keep it in check. However, some moulds are bad for tarantulas and in the presence of a severe prolonged mould outbreak tarantulas can absolutely get book lung infections.
Having said all that, I wouldn't be too concerned about the tarantula harbouring nasties if you do get it. You can quarantine it for the first couple of weeks by keeping it in a separate room to the rest of your collection, and cleaning thoroughly any shared tongs or tools used in very hot water. It's good practice to do that anyway.