When it comes to finding affordable neighborhoods to rent an apartment in Boston, there are a few options to consider. Here are some of the most budget-friendly neighborhoods in the city:

1. Dorchester: Located in the southern part of Boston, Dorchester offers relatively affordable rental options compared to other neighborhoods. It has a diverse community and offers easy access to public transportation.
2. Roxbury: Roxbury is another neighborhood that provides more affordable rental options. It has a rich cultural history and is known for its vibrant arts scene. Roxbury also offers convenient access to downtown Boston.
3. East Boston: Situated across the harbor from downtown Boston, East Boston is an up-and-coming neighborhood with affordable rent prices. It offers beautiful waterfront views and is well-connected to the city through public transportation.
4. Mattapan: Mattapan is a neighborhood located in the southern part of Boston. It offers affordable rental options and a suburban feel. It is well-served by public transportation and has various parks and green spaces.
5. Jamaica Plain: Although Jamaica Plain has seen an increase in rent prices in recent years, it still offers relatively affordable options compared to downtown Boston. It has a vibrant community, plenty of green spaces, and a lively restaurant scene.

It's important to note that rental prices can vary depending on factors such as apartment size, amenities, and location within each neighborhood. It's always recommended to research and compare prices thoroughly before making a decision.

For more information on rental prices and availability, you can check popular real estate websites such as Zillow or Realtor.com. These platforms provide up-to-date listings and can help you find affordable apartments in your desired neighborhood.

Remember to consider your specific needs and preferences when choosing a neighborhood, as affordability is just one aspect to consider when searching for an apartment in Boston.