The reason it is hard to find an apartment in Boston while owning a dog is because the landlord must pay for the insurance on the dog. Even if you have your own renter's insurance ( which you should) it can be very difficult. That being said, the breed, weight and amount of your budget are all parts of finding an apartment. The majority of luxury buildings (price example $2500 1bd) will accept dogs. Also, the South End/Brighton/Somerville are some neighborhoods where dogs seem to be accepted more often.
Honestly, you are better off commuting into the city if you would like to own a dog. And once you find an apartment, NEVER move.
I hope this helps.