In my 30 years experience, it is very safe.
I live in the Martinsville section, north of Rte.22 and south of Rte. 78. I have walked late at night and run 5 miles every morning between 4-5 AM and have never feared being by myself.
You can contact the Bridgewater Police, let them know you are looking to move here and ask them for stats. They are very helpful. You'll need to tell them which section you are interested in moving to.
Bridgewater is a big town, so be specific.
Bridgewater Police Department
100 Commons Way
Bridgewater, NJ 08807
No town is desirable if the crime rate is high. And Bridgewater has always been desirable as long as I can remember.
As for walking distance to shopping, mass transit & parks: Every section of Bridgewater has parks. How far is the walk you are willing to do?
If you want to walk to Mass transit: are you talking bus or train? The bus line runs along Rte. 28 & Rte 22. The train is by the Baseball stadium, which is accross the street from Target.
There is also a section of Bridgewater next to Raritan. Raritan has a train station, as well.
I am not sure where you are looking to move to. We can be more helpful, if you are more specific.