You don't have to drop a lot of dough to enjoy a full day in Boston, food and all. You just have to know where to go to make the money you have last. You can enjoy art, spend the day in a park, picnic along the waterfront, listen to live music, eat at a trendy restaurant, and even see live theater without spending a small fortune.
Transportation is the key ingredient to successful savings in Boston. With garage parking often costing over $30, you can save a small fortune walking around Boston or utilizing the T rather than driving and paying to park. Leave your car behind and make your away around the city like savvy locals do.
If you want to see a show without spending too much pop in to BosTix. Boston has multiple BosTix ticket booth locations, where for 30+ years , they have been a reputable outlet providing discount tickets to many of Boston's shows. BosTix offers in person, 1/2 price, day-of-show tickets to performances. Making BosTix an easy way to save some money and still getting a bit of culture.
For the art enthusiast the Boston Public Library welcomes you to enjoy their rotating exhibits as well as the building itself. Located in Copley Square, this building's extraordinary architecture is a work of art itself. The library hosts many free lectures, as well as workshops and movies. There is a free tour of the building that focuses on the art and architecture within.
Art at the Boston Public Library — Photo courtesy of redjar/flickr
If you're looking to take in some history, you can always walk the Freedom Trail, but if that is biting off more than you can handle, check out the USS Constitution, Docked at the Navy Yard in Charlestown. This historic ship offers free admission and includes a guided tour that takes place every half-hour. The Constitution Museum is also free, with a suggested donation and is located in close proximity to the ship. 
Old Ironsides — Photo courtesy of Jessica Polizzotti
Good food can be found on the cheap, thanks in part to Boston's laws against drink specials during happy hour. Many Boston restaurants have found a way to draw the after work crowd by offering reduced price food. The tapas bar, Estragon, offers a selection of menu items at the bar for $1, $3 or $5 Monday-Thursday from 5-7pm. While high end steakhouse, Morton's serves "Bar Bites" for under $10 Monday - Friday from 4:30pm - 6:30pm and 9pm - Close. Aside from these special promotions, there are plenty of Boston restaurants where you can fill your belly without emptying your wallet, like Silvertone Bar and Grill where their comfort food favorites have crowd pleasing prices and Bova's in the North end provides the masses with stuffed sandwiches, pastries and Sicilian slices to go. Perfect for a picnic at Christopher Columbus Park.