You could stay in Providence, Rhode Island, and take the commuter rail up to Boston for the day.

Or some of the historic towns around Boston that are on the commuter rail might be affordable. But probably not Salem in October, unless it's early October.

There are also campgrounds within range of the T and commuter rail, including Bumpkin Island in Boston Harbor. But October is risky for cold weather - it's not unusual to get snow just before Halloween.

Normandy farms in Foxboro, Mass is 30 minutes south of the city

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I would not recommend camping in New England in October unless you’ve got great gear and have experience cold weather camping.

October is when New Englanders struggle to go another night before turning the heat on in the house.

Despite our complaints, there is a mostly working train system. Maybe look for a hotel further out near a subway or commuter rail line?

There’s hostels downtown

Otherwise Idk, Portland or providence?

If you don’t mind dorm-style accommodations, the HI Hostel near the theater district is super affordable, under $200 for 2 mid-week nights in October.

Vermont at that time of year is going to be beautiful on account of the fall foliage. Camping in October is a little chilly but should be no problem as long as you've got a bag that can handle lows around freezing in the night.

Maine and NH are also going to be lovely in October, and both of those would give you easier access to cities (Boston, Portland) if you want to get some cultural attractions in while you're here.