Had a great stay at the PanAmerican Hostel in Panama! It is clean and safe. Located in Caso Viejo, it is close to many tourist sites. It is not in the safest area, but is closely patrolled with police and is locked up safely with 24/hour front desk service. It is also very cheap, I would recommend buying a two bedroom because it is not much more costly and has a lot more privacy. You may have to walk a little to get to an area with cabs, however Casco Viejo is within walking distance.
Calle 10 y Avenida A, Calle 10a Oeste 8 - 13, Panamá, Panama