Having rented in a couple of places in Pawtucket before we bought a house here, I'd say that there are just a few neighborhoods to avoid, namely the two housing projects (Prospect Heights and Galego Court) and the little Bermuda Triangle around Magill Street which for some reason is a magnet for trouble. That's where Baldwin Elementary is located, and I'd give that immediate vicinity a miss.
The Pleasant View area off Broadway looks rundown, but isn't unsafe, just tends to be poorer. Superficially it looks like an immigrant neighborhood, but scratch the surface and you will find a large population that has lived there for 40-50 years, the kind of people who keep an eye on things and call the police whenever they see something or someone suspicious.
The Woodlawn area around Carver that the OP asked about is nice; a friend of mine used to live not far from there. Woodlawn in general is not bad, except for the above named area around Main/Magill. (Even so, a couple of years ago some punk robbed the little convenience store there which is run by a well-liked immigrant African family, and the whole neighborhood was up in arms, and helped to catch the guy.)
The best areas in Pawtucket are Countryside (bordering Seekonk, MA), Quality Hill, and Oak Hill. Countryside is primarily single family, so you won't really find apartments there. The other two areas are mixed single and multi-family neighborhoods, so rentals exist, but in smaller numbers, and priced a bit higher.
I have found people in Pawtucket to be friendly, so the best advice I would give is to look at a few places, and talk to people living on the street. This is a good time of year, because people tend to be puttering in their yards or porches, or walking along the street (they are mostly good walkable neighborhoods.)
Also, look for an owner-occupied property rather than an anonymous ant farm if you are concerned about safety. For example, an apartment like this one:
Victorian Gem - Utilities Included. (I know the landlord, and they really are good neighbors.)