The DR in general.. and Punta Cana specifically.. is about as safe a tourist destination you are going to find outside the US (and is safer than many US destinations quite frankly).. .there are a significant number of US expats living in the DR perminantly.. the only complaint I have ever heard any of them express has been that if you have a need for high level, extended medical care.. you need to be prepared to come back to the US (cancer treatments, etc..)... other than that, to a man, everyone I know currently living in the DR absolutely loves it..
I have wandered the streets of Santo Domingo and most of the areas surrounding Punta Cana, by myself, at all hours of the day and night, on many occasions.... with never a problem..
You dont want to be stupid.. be aware of your surroundings.. and dont make yourself an obvious target for crime... but go to Punta Cana and have a great time.. thats what it is there for..
Just treat your visit like you would a visit to any major city in the US....
If it looks like a ghetto.. it probably is... dont wander through there at 4AM in a drunken stupor...
If someone looks like they are giving you the "stink eye".. they probably are... dont f*ck with them.. and dont put yourself in a position where they can f*ck with you...
Go "fun" places.. that are frequented by people that want to have fun... this does NOT limit you to traditional tourist destinations... one of the first things I do when I get to a new country is ask around and find out where I can get a really great meal at a resturant that LOCALS eat at.. not somewhere frequented by tourists.... you are most likely going to be recommended an upscale place, in a nice part of town, etc.. (no one that wants to "show off" their country is going to recommend a hot dog stand in the barrio..)... and Ill ask where upscale locals go for fun (are there any concerts in town? what about festivals? etc..).. do this and you get to experience authentic local culture.. instead of the small piece of culture that tourists normally get to see..
Just do the same thing you would do in the US.. and youll be fine... (you wouldnt go to 8 mile in Detroit by yourself to get some "soul food" at midnight.. and you wouldnt go to "Orange Mound" in Memphis to catch a local Rap concert...)...