Boston advertises their police and fire stations as public bathrooms so you could use the fire station in the middle of the North End. There's a BPL branch as well.

Langone Park has one of those $0.25 quarter operated public bathrooms but most of those I've seen are dilapidated and in a state of disrepair.

If you're around the perimeter, of the North End, you can go to North Station (either the station or the Star Market though the station bathroom is nasty), the Boston Public Market near Haymarket, Faneuil Hall, Marriott Long Wharf (inside is much nicer than the outside bathroom).