The public elementary schools really are not as bad as everyone says. My daughter is in the East Side elementary school (MLK) and the principal, teachers, and parents are all very committed, passionate, and engaged. MLK suffers from district-wide cuts in funding for the arts and sciences, but is making it up through very creative supplementation of sponsored programs from external non-profits and volunteers. Additionally the elementary charter school programs - Highlander, Cuffee, TimeSquared, and International are quite strong and offer additional options (if you make it through the lottery/waiting list).
Middle school options start to get a little more dire, that is the weak spot of PPS. The only middle school on the East Side is so undersubscribed from the neighborhood that the district was proposing to change it to an overflow high school, but ran into community opposition and delayed their decision. A lot of things are changing in the Providence Public School system and in RI education in general right now...we're taking our chances that current challenges and heightened awareness will ultimately change it for the better.
My observation is that many East Siders give up on PPS based on anecdotes, rather than experience, and opt early on for the crushing double bill (taxes+tuition) of private schooling. But so much of it is based on your kid's personality and needs.
As for safety, the District (8) that encompasses the East Side is the safest in the city for violent crime, and has had the largest drops in both violent crime and property crime, per a recent neighborhood presentation from the Police Dept. I would feel comfortable riding my bike home at 10 from the train station. Walking would depend on the street you chose, but you could be in for quite a walk uphill from the train station !