Providencia is a nice play to stay. Its more of a upmarket residential area. Its actually a very long stretch of road. Use google map to get a feel of it. U can take metro to go downtown. Find a hotel near metro station. Downtown area is Ok too to stay. Santiago is generally a safe place. Try not to be flashy. Personally I prefer to stay near downtown as its walking distance to major attractions such as Plaza des Armas but Providencia has better ambience when it comes to food.
if u are looking for one with swimming pool at Providencia u can check Hotel Santiago Park Plaza. I previously stayed at Torremayor opposite the much grander Park Plaza. The website shows it has swimming pool.