Civil engineer here. Alot of people are commenting here that this is for drainage, but I seriously doubt that's the case. The cross slope of these sidewalks are wayyyyyy more than is necessary for proper sidewalk drainage. For reference, minimum slope percentage we design for on sidewalks to accommodate drainage towards the road is 0.5%, with a maximum allowed percentage of 2% per ADA requirements. Even at 2% slope, it's barely noticeable when walking. If I'd have to guess, I'd guess these sidewalks have a cross slope of at least 15%+. I doubt it's for sewage either, because although Boston is old, I don't think it's that old.

If you look at the pic, it looks like the grey building face on the right and the sidewalk panels that abut it are newer than the other buildings and sidewalk panels. I bet what happened is they re-did the building which raised the building face up vertically, but didn't take into account the impact it would have on the sidewalk so the City was forced to make the sidewalk with this crazy cross slope to meet the new building face.